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Yol 


../.&... 


I 


—OF- 

H.  T.  CORY,  M.  E.,  C.  E. 

LAFAYETTE,   INDIANA. 


GIFT    OF 

Dean   Cory 


ENGINEERING  r 


.  ELECTRICAL  MEASUREMENT 

A  N  D   T rf EJ  /A  L  H'  0 : V 

GALVANOMETER; 

ITS  CONSTRUCTION  AND  USES, 


By  T.    D.    LOCKWOOD. 


PUBLISHED   BY 

J.     H.    BUNNELL    &    CO., 

No.   112    Liberty   Street. 


NEW    YORK, 

1883. 


-aM 
ENGINEERING  LIBRARY 


Copyrighted  by  J.  H.  BUNNELL  &  Co.,  New  York. 


Press  of  Wm.  H.  Bartholomew  &  Bro.,  "22  College  Place,  New  York. 


RREKACB. 


THE  author  has  made  it  his  aim,  in  the  prepara- 
tion of  this  little  book,  to  set  forth  the  principles  of 
electrical  measurement  and  the  construction  and 
uses  of  the  most  important  forms  of  the  galvano- 
meter in  as  simple  and  concise  a  manner  as  lay  in 
his  power,  and  in  language  as  plain,  as  possible. 

All  who  have  passed  through  the  amateur  stage 
of  electrical  science,  know  that,  irrespective  of  the 
clearness  and  lucidity  of  explanations  made  in  that 
science,  there  remains,  owing  to  the  absolute  mys- 
tery of  the  force  on  which  it  is  based — electricity 
itself — a  certain  darkness  and  ambiguity  which  is 
very  difficult  to  dispel,  and  which,  to  students,  is  in 
the  highest  degree  discouraging.  Yet,  by  bringing 
common  sense,  common  sense  language,  industry 
and  perseverance  to  bear  upon  the  subject,  much  of 
this  darkness  may  be  dissipated. 

There  is  no  branch  of  electrical  science  more 
beautiful,  more  interesting,  and  we  may  even  say 
more  entertaining,  than  electrical  measurement  and 
testing ;  yet  it  is  well  known  that  its  processes  have 
been  contemplated  with  strong  aversion  by  many  to 

844038 


11. 


whom  a  knowledge  of  such  manipulations  would 
prove  invaluable. 

Knowledge  of  the  higher  mathematics  is  such  an 
invaluable  aid  in  these  processes,  that  it  is  not  sur- 
prising that  those  familiar  with  algebra,  and  the 
differential  and  integral  calculus,  should  have  availed 
themselves  extensively  of  such  an  aid  ;  it  is,  however, 
to  be  lamented  that  nearly  all  the  text  books  on  this 
most  important  subject  assume  their  readers  to  be 
proficient  in  mathematical  knowledge,  and  use 
mathematical  symbols  to  explain,  or  rather  to  dis- 
guise, the  most  ordinary  measurements. 

Many  students  are  thus  frightened  away  from 
galvanometrical  tests,  for  just  as  the  school-boy 
dreads  the  processes  of  arithmetic,  and  sings  the  old 
rhyme  : 

"  Multiplication  is  vexation, 
Division  twice  as  bad  ; 
The  rule  of  three  doth  puzzle  me, 
And  practice  drives  me  mad," 

so  the  majority  of  his  full-grown  brethren  regard 
with  suspicion  algebraic  equations  and  symbols, 
whether  easy  or  complicated. 

Whatever  otherwise  may  be  the  faults  of  this  little 
book,  and  doubtless  they  are  many,  it  is  the  proud 
boast  of  both  author  and  publisher  that  no  algebraic 
equation  appears  therein,  and  that  arithmetic  has 
been  found  sufficient  for  the  formulae  contained  in 


111. 


its  pages.  It  is  not  expected  that  the  information 
contained  herein  will  be  greatly  beneficial  to  exper- 
ienced electricians ;  but  it  is  hoped  that  students, 
operators,  inspectors  and  amateurs  will  find  it  an 
assistance  in  their  labors  and  in  their  pursuit  of 
knowledge,  and  that  it  will  measurably  fill  a  long- 
vacant  niche  in  electrical  literature. 

The  best  electrical  text-books  have  been,  in  its 
preparation,  consulted  and  extensively  drawn  upon. 
We  are  especially  indebted  to  Kempe's  "  Electrical 
Testing,"  Haskins'  "  Galvanometer,  and  its  Uses," 
Schwendler's  "  Testing  Instructions,"  and  Thomp- 
son's "  Electricity  and  Magnetism." 


MALDEN,  Mass.,  March,  1883. 


ELECTRICAL  MEASUREMENT  A\D 

ITS 


WHEN  a  man,  intending  to  build  a  house,  a  busi- 
ness block,  or  a  factory,  goes  to  buy  his  land  as  the 
preliminary  transaction,  he  has  usually  calculated 
the  cost,  size  and  character  of  the  buildings  he  pur- 
poses to  erect,  and  he  has  estimated  the  quantity  of 
land  necessary.  Finding  a  lot  which,  in  other  res- 
pects, meets  his  requirements,  he  ascertains  its  size 
by  measuring  ;  for  if  his  intention  is  to  put  up  build- 
ings covering  a  superficial  extent  of  ten  acres,  it  is 
hardly  probable  that  he  will  attempt  to  do  it  on  a 
five-acre  lot.  He  will  not  ordinarily  set  out  to 
build  a  block  having  a  frontage  of  five  hundred  feet 
on  a  land  frontage  of  a  hundred  feet. 

If  we  are  buying  our  winter's  supply  of  coal,  we 
buy  it  by  the  ton,  and  we  want  it  weighed  ;  we  do 
not  care  to  have  the  coal  dealer  guess  at  the  weight, 
and  pay  him  perhaps  for  six  tons  when  we  receive 
but  five.  Similarly,  the  coal  dealer  himself  finds  it 
to  his  interest  to  weigh  his  coal,  as  he  on  his  part 
does  not  care,  as  a  rule,  to  receive  a  cash  equivalent 
for  five  tons,  when  he  has  furnished  six. 

In  carpeting  a  room,  a  person  with  but  a  grain  of 
common  sense,  first  measures  the  room,  and  orders 


accordingly ^a^qrig;  "  I  want  so  many  yards."  He 
wouM  never  think  of  .looking  at  the  room,  and  then 
-waging:  <>ff  l4  the  carpet  store  and  saying,  "Well, 
unroll  your  carpet ;  I'll  tell  you  when  there's 
enough." 

In  commencing  any  branch  of  manufacture  re- 
quiring power,  no  one  would  think  for  a  moment 
of  putting  in  a  hundred  horse-power  steam  engine 
to  do  work  requiring  but  one  horse-power  ;  neither 
would  we  contemplate  the  desirability  of  utilizing  a 
donkey  engine  to  set  a  dozen  quartz  mills  in  motion. 

The  power  of  the  steam  engine  or  other  motor  is 
always  in  some  degree  at  least  proportionate  to  the 
work  to  be  done. 

Thus,  in  all  the  daily  commercial  and  mechanical 
transactions  of  life,  we  are  accustomed  to  institute 
some  system  of  measurement,  weight,  or  compari- 
son, whereby  we  may  intelligently  buy,  sell,  and  use 
the  various  agencies,  necessaries  and  comforts  of 
civilized  life. 

Singularly  enough,  during  the  early  days  of  tele- 
graphy, and,  indeed,  until  a  comparatively  recent 
date,  measurements  in  electricity  were  scarcely 
heeded  or  thought  of,  and  the  chapter  of  accidents 
was,  in  a  great  measure,  trusted  to,  in  the  construc- 
tion and  maintenance  of  a  line  of  telegraphy. 

The  line  was  built,  perhaps,  of  several  gauges  and 
grades  of  wire,  the  ground  plates  consisting  of  two 


3 

or  three  turns  round  a  Irusty  gas-pipe  at  both 
termini,  the  battery  entirely  disproportionate  to  the 
work — perhaps  much  too  large,  perhaps  much  too 
small,  the  relays  varying  from  one  hundred,  to  four 
hundred  ohms,  the  insulation  defective  and  irregular, 
and  the  local  circuits  likely  enough  having  two-ohm 
sounder  magnets  and  six-ohm  batteries. 

In  fact,  while  carrying  the  exact  sciences  into 
every  common  transaction  of  the  day,  we  departed 
from  common  sense  and  exact  measurements  where 
both  elements  were,  if  anywhere,  most  requisite 
with  the  results  that  might  be  expected,  i.  e.,  irre- 
gular, poor-working  lines,  yet  withal  much  higher 
in  first  cost  and  maintenance  than  if  they  had  been 
properly  constructed  on  a  basis  of  electrical  meas- 
urements, thus  insuring  a  proper  proportion  between 
the  different  elements  of  the  line. 

There  is  no  force  of  nature,  however,  that  is  more 
subject  to  her  unchangeable  laws  than  electricity, 
and  many  of  those  laws,  and  the  measurements  and 
testing  methods  depending  thereon,  are  so  exceed- 
ingly simple,  that  a  child  may  readily  master  them. 

Every  telegraph  operator,  every  telephone  man- 
ager or  inspector,  and  every  person  engaged  in  any 
employment  in  which  electricity  is  used,  should  be 
conversant  with  simple  testing  and  measurements. 

Not  only  is  such  knowledge  a  most  useful 
acquirement,  but  its  practice  is  interesting  and 


fascinating  in  the  extreme,  even  if  indulged  in  merely 
as  a  pastime.  Just  as  in  the  cases  and  circumstances 
instanced,  we  employ  weights  and  measures  ;  so  we 
may  in  the  applications  of  electricity,  employ  various 
measurements  for  many  purposes,  and  we  shall  find 
that  by  basing  our  practice  on  these  measurements, 
we  shall  have  both  better  and  cheaper  results. 

As  also  we  commonly  employ  suitable  instru- 
ments and  apparatus  in  weighing  and  measuring 
tangible  substances,  using  scales  or  balances  for 
those  bought  and  sold  by  weight,  tape  measures  and 
rules  for  measurements,  and  clocks  and  watches  to 
mark  the  advance  of  time,  so  we  find  it  essential  in 
the  valuations  and  comparisons  of  electricity  to  use 
suitable  instruments. 

These  instruments  are  called  galvanometers,  and 
are  capable  of  being  used  to  measure,  compare  and 
estimate  many  of  the  different  properties  and  mag- 
nitudes of  electricity.  By  their  aid  we  may  readily 
ascertain  and  compare  the  working  strength  and 
value  of  electric  currents,  the  resistance  which  elec- 
tro-magnets, wires  and  other  conductors  offer  to  the 
passage  of  the  current,  the  electro-motive  force  or 
initial  power,  and  the  resistance  of  batteries  ;  and 
we  are  also  enabled  by  their  use  to  localize  line 
troubles  and  other  circuit  faults. 

As  batteries  are  essential  in  all  electrical  measure- 
ments, they  will  be  now  briefly  referred  to. 


BATTERIES. 

The  most  suitable  battery  for  use  in  connection 
with  galvanometers,  at  least  in  ordinary  measure- 
ments, is  the  blue  vitriol  battery  in  almost  any  of 
its  well-known  forms. 

That  type  known  as  the  Crow  Foot  Battery  is  to 
be  commended  as  a  thoroughly  efficient  form,  and 
is  now  in  general  use  on  telegraph  lines,  both  for 
"  main  "  and  "  local  "  purposes. 

This  battery  is  shown  in  Figure  i,  and,  as  plainly 
indicated,  is  of  the  most  simple  nature ;  consisting 
practically  of  but  three  parts,  i.  e.,  the  zinc,  copper, 
and  containing  vessel. 


Figure  i. — THE  CROW  FOOT  BATTERY. 


The  Leclanche  battery  also  gives  very  fair  satis- 
faction, and  inasmuch  as  'it  is  now  universally 
employed  for  many  purposes,  and  is  well  known  to 
almost  every  one,  it  is  here  noted  as  being  adapted 
for  this  purpose,  and  recommends  itself  as  being 
always  at  hand  or  easily  procurable. 

Both  forms  of  battery  are  so  familiar  to  every 
one  engaged  in  electrical  pursuits,  that  no  descrip- 
tion of  either  is  here  necessary.  A  few  words  on 
the  management  of  batteries  may  not,  however,  be 
out  of  place. 

If  the  Daniell,  or  gravity  battery  be  chosen,  and 
these  are  preferable  on  account  of  their  greater  con- 
stancy, the  best  quality  of  blue  vitriol  ought  always 
to  be  used. 

Never  use  porous  cups  after  they  are  at  all 
cracked,  or  in  any  way  damaged,  or  let  the  zinc 
touch  the  porous  cup.  The  zinc  solution  is  at  its 
best  when  it  is  half  saturated  ;  when  it  is  stronger 
than  that  point  of  saturation,  a  portion  of  the  fluid 
should  be  drawn  off  and  the  cell  filled  up  with 
water. 

When  a  gravity  battery  is  first  set  up,  the  poles 
ought  to  be  united  by  a  wire  for  a  day  or  two  ;  this 
will  tend  to  separate  the  solutions  and  to  concen- 
trate the  zinc  solution. 

Keep  the  level  of  the  water  at  least  a  quarter  of 
an  inch  above  the  zinc.  The  copper  plate  ought 


71 

. 
always  to  be  immersed  in  the  blue  solution,  and  the 

zinc  in  the  sulphate  of  zinc  solution. 

The  line  between  the  two  solutions  should  be 
kept  as  sharp  as  possible.  If  the  blue  is  too  low,  a 
little  of  the  zinc  solution  should  be  withdrawn  and 
replaced  with  pure  water,  and  the  battery  circuit 
left  open  for  a  while. 

If  the  blue  is  too  high,  the  battery  ought  to  be 
short  circuited  by  uniting  its  poles  with  a  wire  for 
a  while. 

If  a  froth  generates  on  the  surface,  it  may  be  re- 
moved with  a  piece  of  wood  or  a  brush. 

When  the  zincs  become  dirty,  they  should  be 
taken  out,  scraped  and  washed. 

When  the  Leclanche  battery  is  employed,  the 
following  directions  may  profitably  be  complied 
with :  Never  let  the  sal-ammoniac  solution  rise 
above  the  shoulder  of  the  jar.  If  a  new  battery  is 
set  up  and  wanted  at  once,  a  little  water  should  be 
poured  into  the  porous  cup.  The  sal  ammoniac 
solution  should  be  strong,  but  too  much  sal- 
ammoniac  should  not  be  put  into  the  jar  at  once,  as 
it  is  likely  to  cake,  instead  of  dissolving. 

If,  on  the  contrary,  there  is  not  enough  in  the  jar, 
crystals  will  commence  to  form  on  the  zinc,  and  the 
power  of  the  battery  will  be  impaired. 

The  connecting  wires  of  a  Leclanche  battery 
should  be  occasionally  looked  over,  as  the  free 


ammonia  generated  by  the  cell  is  likely  to  eat  them 
through. 

If  a  battery  consisting  of  a  number  of  cells  is 
weak,  and  no  cause  is  apparent,  each  cell  should  be 
tested  separately,  and  when  the  defective  cell  is 
found  and  examined,  the  lead  cap  surmounting  the 
carbon  will  probably  be  found  to  be  insulated  from 
the  carbon  by  a  salt  of  lead  formed  under  it. 

If,  by  any  accident,  the  Leclanche  battery  cell  be 
left  on  closed  circuit,  and  run  down,  its  strength 
may  be  to  a  certain  extent  renewed  by  soaking  the 
porous  cups  in  water,  or  dilute  muriatic  acid,  and 
giving  the  battery  a  considerable  rest. 

In  all  batteries,  to  produce  good  work,  every 
point  of  contact  and  every  connection  should  be 
kept  clean  and  bright,  and  every  screw  well  tight- 
ened up. 

It  is  found  convenient  to  illustrate  a  battery  cell 
by  the  symbol  or  conventional  sign  of  a  thick  and 
thin  line  of  different  lengths,  as  in  Fig.  2,  a  number 
of  cells  being  similarly  represented  by  a  series  of 
these,  as  shown  in  the  same  figure. 


llllljl. 

Figure  2. 


I'hy 


The  short,  thick  lines  are  usually  intended  for  the 
zinc,  and  the  longer,  thin  lines  for  the  other  plates. 

The  plus  sign,  -}-,  at  the  end  of  the  battery  desig- 
nated by  the  thin  line  denotes  the  positive,  and  the 
minus  sign,  — ,  at  the  other  end  the  negative  pole. 

DEFINITIONS    OF    ELECTRICAL  TERMS  AND  UNITS. 

In  order  that  we  may  clearly  understand  the  gal- 
vanometer, and  when,  why,  and  how  to  use  it,  it  is 
proper  that  we  should  have,  at  the  outset,  a  full 
comprehension  of  the  meaning  of  the  technical 
terms  commonly  used  to  express  the  different  prop- 
erties, magnitudes,  functions  and  relations  of  elec- 
tricity and  electrical  conductors,  and  of  the  units 
which  indicate  the  value  of  such  properties. 

ELECTRO-MOTIVE    FORCE. 

Electro-motive  force  is  the  name  given  to  the 
initial  power  of  any  source  of  electricity.  If  the 
force  originates  in  a  battery,  it  is  produced  by  the 
difference  in  chemical  action  on  two  metals  in  a 
liquid. 

It  expresses  for  electricity  what  the  pressure  in  a 
boiler  does  for  steam,  and  would  express  the  strength 
of  the  current  if  a  circuit  could  be  made  having  no 
resistance. 

But,  as  in  the  case  of  steam,  much  of  the  power 
is  lost  by  friction  in  the  pipes  and  by  radiation 


10 

between  the  boiler  and  the  cylinder  of  the  engine,  so 
in  electricity,  the  useful  effect  of  the  electro-motive 
force  is  greatly  diminished  by  having  to  overcome 
the  resistance  both  of  the  source  itself  and  of  the 
conductors  it  traverses.  The  electro-motive  force 
of  any  battery,  then,  may  be  defined  as  "  the  power 
which  it  has  to  transmit  a  current  against  resistance," 
or  its  power  to  overcome  resistance. 

It  increases  in  direct  proportion  to  the  number 
of  cells  employed,  ten  cells  having  exactly  ten  times 
the  electro-motive  force  of  one  cell. 

It  is  not,  however,  dependent  on  their  size ;  a  cell 
no  larger  than  the  bowl  of  a  tobacco  pipe  possesses 
as  great  an  electro-motive  force  as  a  cell  with  the 
\J  same  kind  of  plates  and  liquids  which  would  hold  a 
gallon.  If  we  have  to  work  a  long  circuit,  or  one 
of  high  resistance,  we  require  a  strong  initial  force, 
and  we  get  it  by  increasing  the  number  of  cells.  It 
is  convenient  to  use  the  abbreviation  E  M  F  to  ex- 
press the  term  electro-motive  force. 

POTENTIAL. 

Electrical  potential  is  to  electricity  just  what 
temperature  is  to  heat,  and  level  or  height  to  water. 

The  word  "potential,"  literally,  means  the  power 
of  doing  work  ;  electrical  potential  implies  some- 
thing more  ;  it  indicates  the  electrical  condition  of 
any  body,  and  not  only  that  electricity  has  power  to 


11 

do  work,  but  also  thajt  it  is  in  a  condition  to  exer- 
cise that  power. 

The  word  "tension"  is  frequently  used  by  the 
older  writers  upon  electricity,  and  has  substantially 
the  same  meaning  as  "  potential,"  i.  e.,  a  condition  of 
readiness  to  do  work,  such  as  that  of  a  bow,  during 
the  moment  when  the  string  is  drawn  as  tense  as  it 
can  be,  just  before  the  release  of  the  arrow. 

Comparing  electricity  with  heat,  it  is  perfectly 
clear  that  to  transfer  heat  from  one  point  to  another, 
it  is  first  necessary  that  the  two  points  shall  be  of 
different  temperature.  Just  so  with  electricity. 
To  nroduce  a  current,  it  is  requisite  that  the  two 
poipts  which  are  united  by  any  conductor  shall 
be  of  different  potentials,  and  when  such  is  the 
case,  electricity  flows  from  the  higher  to  the  lower 
potential. 

POTENTIAL,   OR  TENSION,   AT  ANY   POINT. 

The  electrical  potential  of  the  earth  is  assumed  to 
be  zero.  When  we  speak  of  the  electrical  potential 
of  any  point  in  a  voltaic  circuit,  for  example,  an 
ordinary  telegraph  line,  we  mean  the  difference  in 
electrical  condition  between  that  point  and  the  zero 
point. 

To  illustrate  the  idea,  let  us  imagine  that  we  de- 
sire to  connect  telegraphically  two  points,  ten  miles 
apart,  and  that  our  electrical  power  consists  of  ten 


12 

cells  of  battery  connected  up  in  the  usual  way ;  that 
is,  in  series,  or  one  after  another. 

If  we  construct  a  single  line  and  make  a  ground 
circuit,  connecting  one  end  of  the  line  with  the 
earth,  the  other  end  of  the  line  with  the  positive 
pole  of  the  battery,  and  the  negative  pole  of  the 
battery  with  the  ground,  we  now  have  a  complete 
circuit.  The  negative  pole  being  united  to  the 
earth,  has  the  same  potential  as  the  earth ;  i.  e.,  zero. 

The  distant  end  of  the  line  being  also  united  to 
the  earth,  has  also  a  zero  potential.  Let  us  further 
suppose  the  EMF  of  every  cell  to  have  the  value  of 
ten  units  ;  this  gives  for  the  ten  cells  a  total  EMF, 
or  difference  of  potential  between  the  two  extreme 
poles  of  the  battery,  of  ten  times  ten,  or  100,  which 
then  must  be  the  potential  of  the  circuit  at  the 
positive  pole,  where  the  line  is  united  to  it ;  and 
because  the  positive  pole  is  to  line,  the  potential  is 
100  -[-,  or  above  zero. 

The  potential  now,  at  any  point  in  the  circuit,  is 
ascertained  by  dividing  the  total  resistance  into  100 
imaginary  equal  parts,  through  which  the  potential 
gradually  falls,  till  it  reaches  zero  at  the  distant  end. 

Thus,  in  the  case  assumed,  where  the  potential  at 
the  junction  between  the  battery  and  the  line  is 
100,  that  point  in  the  line  where  its  resistance  is 
exactly  halved  will  have  a  potential  of  50 ;  and  if 
we  divide  the  total  resistance  into  quarters,  the 


13 


point  denoting  the  quarter  line  nearest  the  battery 
would  have  a  potential  of  75  ;  and  that  nearest  the 
distant  end  of  the  line  a  potential  of  25,  as  shown 
in  the  diagram  Figure  3,  in  which  the  sloped  line 


Figure  3. 

denotes  the  gradual  fall  of  potential  from  the  bat- 
tery end,  where  it  is  highest,  to  the  distant  ground 
end,  where  it  is  lowest. 

If,  on  the  contrary,  we  reverse  the  battery,  unit- 
ing the  negative  pole  to  the  line  and  the  positive  to 
the  earth,  the  positive  pole  would  acquire  the  zero 
potential  of  the  earth,  and  the  negative  pole  a  poten- 
tial of  100  below  that  of  the  earth  ;  and  in  this  case 
the  potential  rises  as  the  distant  end  is  approached, 
and  will,  at  the  junction  of  the  battery  with  the 
line,  be  100  — ,  or  below  the  potential  of  the  earth ; 
at  the  first  quarter  of  the  distance  75  below,  at 
the  middle  of  the  line  50  below,  and  at  the  last 


14 


quarter  25  below  the  zero  of  the  earth,  as  shown 
in  Figure  4. 

In   this   case,    the   line   of   slope  represents  the 
gradual  rise  of  potential. 


Figure  4. 

If  we  have  a  battery  at  both  ends  of  the  line,  of 
course  the  battery  at  one  end  will  have  the  positive 
pole  to  line,  and  that  at  the  other  will  have  the 
negative  pole  to  line. 

In  that  case,  still  assuming  that  each  battery  has 
ten  cells,  each  cell  giving  an  E  M  F,  or  difference 
of  potential  of  ten,  the  entire  difference  of  potential 
between  the  battery  at  one  terminal  station  and  that 
at  the  other  will  be  200,  as  illustrated  in  Fig.  5  ;  for 
as  we  see  the  potential  at  the  ground  plate  of  Station 
A  is  zero,  being  in  contact  with  the  earth,  from  that 
point  to  the  positive  pole  a  it  rises  to  100  -}-,  going 


15 


*w 

out  to  line,  the  potential  falls  regularly  to  a  point 
where  the  resistance  of  the  circuit  is  exactly  halved, 
and  at  that  point  it  again  reaches  zero. 

Continuing  to  fall,  at  the  point  b,  where  the  nega- 
tive pole  of  the  battery  is  united  with  the  line,  it  is 
100  — .  Here  it  commences  again  to  rise,  and  at 
the  ground  end  of  this  battery  it  is  again  zero. 


100  4 


100  — 


Figure  5. 

Suppose  now,  that  instead  of  using  the  earth  as  a 
return  circuit,  we  unite  the  poles  of  a  battery  by  a 
complete  metallic  wire,  as  in  Fig.  6.  We  may  now, 
as  we  have  no  earth  to  regard  as  a  zero  point,  con- 
sider the  center  of  our  battery  to  be  the  zero  ;  then 
the  positive  pole  will  be  +  and  the  negative  — ,  and 
if,  as  before,  we  assume  the  difference  of  potential 
between  the  poles  to  be  100,  the  potential  at  the 
positive  will  be  50  +,  and  that  at  the  negative  pole 
50  — ,  and  the  potential  will  fall  regularly  along  the 
line  from  the  positive  pole,  and  rise  regularly  from 


16 


the    negative    pole,   until  they  both   meet   in    the 
middle  of  the  wire,  where  the  potential  will  be  zero. 


0- 


25 


25- 

-        I 

50- 


Figure  6. 

It  is  immaterial,  so  far  as  regards  the  energy  or 
strength  of  the  electricity,  whether  the  potential  at 
the  point  where  the  line  is  joined  to  the  battery  be 
so  many  units  above  or  below  the  zero  ;  that  is  de- 
pendent on  the  greater  or  less  difference  between 
the  points  of  highest  and  lowest  potential  in  the 
circuit,  and  not  in  the  least  upon  which  side  of  the 
zero  point  that  difference  extends. 

This  condition,  however,  determines  the  direction 
of  the  current,  because  electricity  always  moves 
from  a  higher  to  a  lower  potential. 

We  have  thus,  at  some  length,  endeavoured  to 
make  these  ideas  clear,  because  the  Wheatstone 
Bridge  system  of  measurement — the  most  beautiful 
and  convenient  system  ever  devised — depends  en- 
tirely upon  the  principles  here  enunciated. 


CURRENT    STRENGTH. 

We  use  the  word  "  current "  to  denote  a  contin- 
uous flow  of  electricity.  The  strength  of  current  is 
the  working  power  of  the  electricity  after  it  has  over- 
come the  constant  resistance  of  the  circuit,  or  the 
amount  of  electricity  actually  traversing  the  circuit. 

Since  the  current  is  the  working  result  of  the 
electro-motive  force  divided  by  the  resistance,  its 
strength  may  with  propriety  be  defined  as  the 
amount  of  electricity  realized.  It  depends  partly 
upon  the  E  M  F  of  the  battery,  and  partly  upon  the 
resistance  of  the  circuit. 

When  any  battery  is  joined  up  in  a  closed  circuit, 
the  strength  of  the  current  is  always  equal  to  the 
electro-motive  force  of  the  battery,  divided  by  the 
total  resistance  of  the  circuit. 

This  is  the  simplest  and,  at  the  same  time,  the 
most  important  law  of  the  electric  current.  It  is 
called  Ohm's  Law,  because  the  German  mathema- 
tician, Professor  G.  S.  Ohm,  first  announced  it. 

We  may  illustrate  the  difference  between  the 
E  M  F  of  the  battery,  and  the  strength  of  current 
flowing  in  the  circuit,  in  the  following  manner  :  A 
battery  has  an  EM  F  of  100,  we  join  it  up  in  a 
circuit  having  a  total  resistance  of  50  ohms  ;  the 
strength  of  current,  therefore,  is  100  divided  by  50, 
which  is  2. 


18 

We  shall  presently  find  occasion  to  re-state  Ohm's 
Law  in  somewhat  different  and  more  definite  terms. 

RESISTANCE 

Is  the  name  given  to  the  obstruction  or  opposi- 
tion to  the  passage  of  electricity  offered  by  the  sub- 
stance of  the  circuit  through  which  it  passes. 

Every  substance  offers  some  resistance,  and,  in 
fact,  it  is  the  different  degrees  of  resistance  offered 
by  substances  that  determines  their  division  into 
conductors  and  insulators.  Conductors  have  low, 
and  insulators  very  high  resistance.  Some  metals 
also  have  a  much  higher  resistance  than  others. 

A  conductor  having  a  high  resistance  does  not  let 
the  electricity  pass  so  freely  as  it  would  in  a  con- 
ductor of  low  resistance ;  that  is,  the  quantity 
passing  in  a  given  time  is  diminished. 

The  resistance  of  a  battery  circuit  is  made  up  of 
the  resistance  of  its  several  parts,  i.  e.,  the  wires,  the 
instruments,  and  the  battery. 

The  two  former  we  call  the  external,  and  the 
latter  the  internal  resistance. 

The  internal  resistance  of  a  battery  consists  in  the 
resistance  of  the  liquids  and  of  the  porous  cell,  if 
one  is  used.  It  increases  in  direct  proportion  to 
the  number  of  cells  used  ;  that  is,  if  one  cell  has  an 
internal  resistance  of  one  unit,  a  battery  of  ten 
similar  cells  will  have  a  resistance  of  ten  units. 


19 

. 

The  resistance  of  any  wire  increases  in  proportion 
to  its  length.  If  the  resistance  of  a  mile  of  wire  be 
ten  units,  that  of  fifty  miles  will  be  fifty  times  ten, 
or  five  hundred  units. 

The  resistance  of  any  wire  is  inversely  propor- 
tional to  the  area  of  its  cross  section,  that  is,  the 
resistance  of  a  wire  increases  as  its  weight  decreases. 

JOINT    RESISTANCE. 

If  a  circuit  divides,  as  in  Fig.  7  at  A,  into  two 
branches  which  meet  again  at  B,  the  current  of  elec- 
tricity passing  through  the  circuit  will  also  divide, 


- 

Figure  7. 

part  flowing  through  one  branch,  part  through  the 
other.  The  strength  of  current  in  the  two  branches 
will  be  inversely  proportional  to  their  several  resist- 
ances. 

Thus,  if  a  current  with  a  value  of  50  be  flowing 
in  the  main  circuit  before  dividing,  and  if  the  branch 
circuits  i  and  2  be  equal,  half  of  the  current  will 
flow  through  the  wire  i,  and  half  through  the  wire 
2,  the  current  in  each  branch  having  a  value  of  25. 


20 

Or,  if  the  resistance  of  branch  i  is  three  times  that 
of  2,  one-fourth  of  the  current  will  pass  through 
branch  i  and  three-fourths  through  branch  2.  The 
joint  resistance  of  the  divided  circuit  will  be  less 
than  the  resistance  of  either  branch  considered  alone, 
because  the  conditions  are  exactly  the  same  as  if  the 
one  wire  was  taken  off  and  a  larger  one  substituted 
therefor,  with  a  conductivity  equal  to  the  two  wires. 
Therefore,  we  see  that  the  term  " joint  resistance" 
means  literally  the  resistances  of  two  or  more  wires 
treated  as  one. 

If  we  add  a  third  wire,  making  three  branch  cir- 
cuits, and  the  resistances  of  all  are  equal,  the  joint 
resistance  is  now  but  one  third  of  the  original  wire, 
and  the  conductivity  is  increased  three-fold. 

To  find  the  joint  resistance  of  two  or  more 
parallel  circuits  when  the  resistances  are  equal,  we 
may  choose  either  of  several  methods.  One  way  is 
to  divide  the  resistance  of  one  wire  by  the  number 
of  wires.  For  example  :  5  wires  each  have  a  resist- 
ance of  60  ohms ;  to  obtain  the  joint  resistance  of 
the  5  wires,  we  divide  the  60  by  5,  and  the  quotient 
being  12,  12  is  the  joint  resistance  required. 

Or  we  may,  if  there  are  only  two  wires,  divide 
the  product  of  the  respective  resistances  by  their 
sum.  To  illustrate  :  Two  wires  each  have  the  same 
resistance — 60  ohms  ;  60  multiplied  by  60  equals 
3600,  and  60  plus  60  is  120  ;  then  dividing  3600  by 


21 

i2o,  we  have  as  a  "quotient  30  ohms,  which  is  the 
required  joint  resistance. 

Or  we  may  divide  the  sum  of  the  resistances  by 
the  square  of  the  number  of  the  circuits,  thus  :  6 
circuits  have  a  resistance  of  60  ohms  each  ;  the  sum 
of  these  resistances  is,  of  course,  6  times  60,  or  360, 
and  the  square  of  the  number  of  circuits,  that  is,  6 
multiplied  by  6,  is  36.  Dividing  360  by  36,  we  find 
the  result  to  be  10,  which  is  obviously  the  joint 
resistance  of  the  6  circuits. 

When  the  resistances  of  the  circuits  are  unequal, 
the  following  plan  must  be  adopted  :  If  only  two 
wires,  we  may  divide  the  product  of  the  resistances 
by  their  sum  as  before.  If  the  joint  resistance  of 
more  than  two  circuits  be  required,  first  find  the 
joint  resistance  of  any  two  of  them,  then,  consider- 
ing this  as  one  resistance,  combine  it  with  a  third, 
and  so  on. 

Let  us  suppose  that  three  wires  have  respectively 
the  following  resistances  :  200,  300,  and  100  ohm's  ; 
we  first  take  two  of  them  ;  200  and  300  multiplied 
together  is  equal  to  60,000  ;  200  plus  300  is  500  ; 
dividing  60,000  by  500,  we  find  the  quotient  to  be 
1 20  ohms,  which  is  the  joint  resistance  of  the  first 
two  wires. 

Calling  that,  now,  one  circuit,  we  multiply  120 
by  100,  the  resistance  of  the  third  wire,  finding  the 
product  to  be  12,000  ;  120  added  to  100  being  220, 


22 

which  gives  us  as  a  result  54  and  a  fraction,  this 
being  the  joint  resistance  of  three  circuits. 

UNITS    OF    ELECTRICAL    MEASUREMENT. 

We  have  now  seen  that  the  batteries  by  which 
electricity  is  developed,  the  conductors  by  which  it 
is  transferred,  the  instruments  by  which  it  is  made 
useful,  and  the  electric  current  itself,  have  certain 
properties,  magnitudes,  or  qualities,  which  it  is  often 
necessary  to  measure  in  order  that  their  working 
value  may  be  properly  estimated. 

That  we  may  be  able  to  make  such  measurements 
and  to  state  their  results,  it  is  essential  that  we  have 
some  standard  terms,  or  units,  which,  when  ex- 
pressed, convey  to  the  mind  definite  ideas,  precisely 
as  in  measuring  a  distance  we  would  say  so  many 
feet  or  miles ;  or  in  expressing  the  flow  of  water,  so 
many  gallons  per  minute  ;  or  in  describing  the  con- 
tents of  a  solid  block,  as  so  many  cubic  feet. 

Furthermore,  when  one  substance  has  several 
properties  or  magnitudes,  a  different  system  of 
measurement  is  required  for  each  magnitude ;  for  as 
in  a  cubic  block  of  wood  we  should  measure  one  of 
its  sides  by  superficial  measure,  its  contents  by  cubic 
measure,  and  its  weight  by  still  another  system,  and 
would  state  the  result  differently  in  each  case,  so 
the  different  electrical  magnitudes  each  have  their 
own  units  in  which  the  results  are  expressed. 


23 

Sometimes  we  find  that  the  results  of  certain 
measurements  are  obtained  by  reference  to  several 
different  magnitudes  ;  as,  for  example,  when  we  time 
the  speed  of  a  horse,  or  a  locomotive,  we  take  the 
length  of  the  distance  traversed  and  the  time  con- 
sumed in  traveling  that  distance,  and  thus  calculate 
the  velocity  by  reference  to  both  length  and  time. 
In  certain  electrical  measurements  we  find  it  neces- 
sary to  resort  to  the  same  process,  and  combine 
different  units  to  obtain  a  definite  result. 

Designations  have  been  given  to  the  practical 
electrical  units  from  the  names  of  distinguished 
electricians  and  scientists.  The  unit  of  electro- 
motive force  is  called  the  "volt,"  from  Volta ;  and 
the  unit  of  resistance  is  called  the  "ohm,"  after 
Ohm,  the  German  physicist  and  mathematician  ; 
while  the  unit  of  current  strength  is  called  the 
Ampere,  from  the  French  philosopher  of  that  name. 

THE  VOLT. 

The  "volt"  is  the  unit  of  E  M  F,  and  has  very 
nearly  the  same  value  as  a  single  cell  of  Daniell 
battery.  Its  precise  value  is  .9268  of  a  Daniell  cell 
in  good  condition  ;  in  other  words,  the  Daniell  cell 
is  equal  in  E  M  F  to  one  volt  and  seventy-nine 
thousandths  —  i  .079. 

The  volt  is  equivalent  to  the  electro-motive  force 
required  to  produce  a  current  of  the  strength  of  one 


24 

ampere  in  a  circuit  having  a  total  resistance  of  one 
ohm. 

The  electro-motive  force  of  most  of  the  gravity 
batteries  is  almost  the  same  as  that  of  the 
Daniell,  and  the  E  M  F  of  the  Leclanche  cell,  is 
1.481,  or  one  volt,  and  four  hundred  and  eighty-one 
thousandths. 

THE    OHM. 

The  standard  unit  of  resistance,  which  we  call  the 
"  ohm,"  may  be  defined  as  a  resistance  about  equal 
to  that  offered  by  a  wire  of  pure  copper,  one  twen- 
tieth of  an  inch  in  diameter  and  two  hundred  and 
fifty  feet  long.  Or  it  may  be  compared  to  one 
sixteenth  of  a  mile  of  No.  9  galvanized  iron  wire  ; 
or  to  one  mile  of  copper  wire,  No.  4;^  Birmingham 
wire  gauge,  which  is  twenty-three  hundredths  of  an 
inch,  or  nearly  a  quarter  of  an  inch  in  diameter.  It 
is  also  approximately  equal  to  a  piece  of  No.  35 
copper  wire,  between  7  and  8  feet  long.  A  mile  of 
No.  12  galvanized  iron  wire  has  an  average  resist- 
ance of  about  32  ohms. 

The  mark  which  may  usually  le  found  stamped 
on  the  base  of  a  relay  denotes  the  resistance  of  the 
coils  from  one  binding  screw  to  the  other.  If,  for 
example,  we  have  a  relay  marked  100  ohms,  we 
know  that  that  is  the  measured  resistance  of  the 


two  spools,  and  that  Hit  is  equal  to  about  three  miles 
of  No.  12  galvanized  iron  wire. 

THE    AMPERE. 

The  unit  of  current  strength  was,  until  very  lately, 
called  a  "weber,"  but  is  now  called  the  Ampere; 
it  may  be  defined  as  the  strength  of  a  current  pro- 
duced in  a  circuit  having  a  total  resistance  of  one 
ohm  by  an  electro-motive  force  of  one  volt. 

If,  for  example,  we  have  a  circuit  consisting  of 
one  cell  of  battery,  an  electro-magnet,  and  the 
necessary  connecting  wires,  as  in  the  diagram 
Fig.  8,  the  battery,  we  will  suppose,  having  an 


Figure  8. 

E  M  F  of  one  volt  and  an  internal  resistance  of 
one  third  of  an  ohm,  the  electro-magnet  and  con- 
necting wires  also  have  resistances  of  one  third  of 
an  ohm  each,  making  a  total  resistance  of  one  ohm 
in  circuit. 

The  current  flowing  in  this  circuit  will  have 
a  strength  of  one  ampere.  A  mille-ampere  is 
one  thousandth  of  an  ampere,  and  is  made  use 
of  in  computing  currents  of  comparatively  feeble 
strength. 


OHM  S    LAW. 

We  rnay  now  re-state  Ohm's  Law,  giving  specific 
values  ;  thus  : 

The  strength  of  current  in  amperes  flowing- 
through  a  circuit  is  equal  to  the  number  of  volts  of 
electro-motive  force  divided  by  the  number  of  ohms 
of  resistance  in  the  entire  circuit. 

The  strength  of  current  is  ascertained  by  taking 
the  E  M  F  in  volts,  and  dividing  that  number  by 
the  total  resistance  of  the  circuit,  including  that  of 
the  battery,  wires,  and  instruments  in  ohms.  The 
result  will  be  in  amperes,  or  fractions  thereof. 

Let  us  see  how  this  works  out.  A  battery  having 
an  E  M  F  of  50  volts,  and  an  internal  resistance  of 
75  ohms,  is  connected  in  circuit  with  a  galvanometer 
having  a  resistance  also  of  75  ohms,  and  connecting 
wires  having  a  resistance  of  10  ohms.  The  total 
resistance  in  the  circuit  is  that  of  the  battery,  gal- 
vanometer, and  wire  added  together,  i.  e.,  160  ohms. 
To  find  the  strength  of  current,  we  divide  the  50 
volts  by  the  160  ohms,  which  gives  us  a  quotient  of 
.3125  of  an  ampere,  or  312^  milli-amperes. 

Therefore  if  we  know  the  E  M  F  and  resistance 
of  any  circuit,  we  can  easily  figure  out  the  strength 
of  current.  On  the  same  principle,  knowing  the 
E  M  F  in  volts,  of  a  battery  and  the  current  in  am- 
peres produced  thereby  in  a  given  circuit,  we  can 


27     , 

ascertain  the  resistances  f  that  circuit,  including  that 
of  the  battery,  by  dividing  the  E  M  F  by  the  cur- 
rent. Likewise,  the  value  of  the  E  M  F  may  be 
obtained  if  we  know  that  of  the  current,  and  of  the 
total  resistance  of  the  circuit,  for  if  we  multiply  the 
resistance  in  ohms  by  the  current  strength  in  am- 
peres, we  find  the  value  of  the  E  M  F  in  volts. 


THE  GALVANOMETER. 

The  galvanometer  is  an  instrument  for  indicating 
the  presence  and  direction  of  currents  of  electricity, 
and  for  measuring  their  strength. 

One  of  its  most  important  functions  is  the  testing 
and  measurement  of  the  resistance  of  line  wires, 
instrument  coils,  batteries,  and  insulation.  It  is 
also  employed  in  detecting  and  localizing  circuit 
troubles  in  telegraph  and  telephone  lines,  and  in 
some  cases  such,  for  example,  as  the  Atlantic  cables, 
as  a  receiving  instrument  for  telegraphic  signals. 

Its  operation  depends  upon  the  action  of  the  two 
forces — electricity  and  magnetism, — and,  though 
galvanometers  are  made  in  many  forms  and  are  used 
in  several  different  ways,  they  are  all  based  on  the 
fundamental  fact  that  a  magnetic  needle  is  deflected, 


28 

or  turned  aside,  from  its  natural  position  by  the 
passage  of  a  current  of  electricity  in  a  conductor 
placed  parallel  to  it. 

When  a  steel  needle  is  magnetized,  and  delicately 
pivoted  at  its  centre,  so  that  it  is  free  to  move  hori- 
zontally, every  one  knows  that  it  will  set  itself  north 
and  south  ;  a  common  example  being  the  ordinary 
compass  needle. 

This  action  of  the  needle  is  due  to  the  influence 
of  the  earth,  which  is  itself  an  enormously  large  and 
strong  magnet.  All  magnets  attract  the  opposite 
poles,  and  repel  the  similar  poles  of  other  magnets, 
and  thus  the  north  pole  of  the  earth  attracts  the 
south  pole  of  the  magnetic  needle,  causing  the 
needle  to  point  north  and  south  as  it  does. 

It  must  not  be  forgotten  that  although  we  call 
the  north  pointing  pole  of  the  needle,  the  north  pole, 
it  is  certainly  the  pole  of  opposite  character  to  the 
north  pole  of  the  earth,  and  therefore,  speaking  cor- 
rectly, we  should  call  it  the  south  pole,  and  the 
French  do  so  call  it.  We,  however,  have  become 
accustomed  to  call  it  "  north,"  simply  because  it 
points  to,  or  seeks  the  north  ;  and  we  shall  proba- 
bly continue  so  to  call  it,  being  a  people  of  steady 
habits.  It  will  be  properly  expressed  if  we  strike  a 
mean  between  the  two,  and  call  it  the  north-seeking 
or  north-pointing  pole.  Of  course,  these  remarks 
may  be  equally  applied  to  the  south-seeking  pole. 


29 

Jf  we  forcibly  turn  the  needle  from  its  north  and 
south  position  to  a  new  position,  pointing  east  and 
west,  as  soon  as  we  remove  the  force,  back  it  goes 
to  the  north  and  south  position  again. 

Thus  we  see  that  the  magnetism  of  the  earth 
exercises  a  constant  force  on  the  needle,  tending  to 
keep  it  pointing  north  and  south. 

In  1802,  Romagnesi,  a  physicist  of  Trente,  in 
Italy,  discovered  that  a  current  of  electricity  affects 
a  magnetized  needle,  and  causes  it  to  turn  from  its 
usual  position.  He,  however,  took  no  great  pains 
to  publish  his  discovery,  and  it  was  neglected  and 
soon  dropped  into  oblivion. 

In  1819,  Oersted,  of  Copenhagen,  ascertained 
that  if  a  wire  conveying  an  electric  current,  be  placed 
parallel  to  a  magnetic  needle,  the  needle  is  drawn 
away  from  its  position  pointing  north  and  south, 
and  tends  to  set  itself  in  a  new  position  at  right  an- 
gles to  the  electric  wire. 

The  amount  of  deflection  depends  to  a  certain 
extent  upon  the  strength  of  the  current,  and  the 
direction  of  the  deflection  depends  upon  the  direc- 
tion of  the  current.  If  the  electric  wire  is  above 
the  needle,  and  the  direction  of  the  current  is  from 
north  to  south,  the  needle  will  tend  to  point  east- 
wardly.  Leaving  the  wire  still  above  the  needle, 
and  changing  the  direction  of  the  current  so  that 
now  it  flows  from  south  to  north,  we  find  that  the 


30 

north  end  of  the  needle  now  deflects  in  a  western 
direction. 

If  we  now  change  the  wire  to  a  position  under 
the  needle,  we  find  all  the  motions  to  be  reversed  ; 
for  passing  a  current  from  north  to  south,  the  needle 
now  exhibits  an  inclination  to  turn  to  the  west, 
while  if  we  pass  the  current  from  south  to  north, 
the  needle  has  an  eastward  inclination. 

It  should  be  here  explained  that  when  we  speak 
of  the  deflection  of  a  needle,  the  north  end  of  the 
needle  is  uniformly  the  one  referred  to,  the  south 
end  of  course  moving  oppositely. 

We  may  readily  understand  the  reason  for  these 
movements,  and  why  they  should  occur.  We  have 
already  indicated  the  cause  of  the  natural  inclination 
of  the  magnetized  needle  to  place  itself  in  a  position 
pointing  north  and  south,  to  be  the  attraction  of  a 
much  stronger  magnet — the  earth, — and  we  may 
easily  believe  that  an  unseen  force  which  causes  the 
needle  to  point  away  from  the  north,  must  also  be 
of  a  magnetic  character  ;  and  so  it  proves  to  be,  and 
the  reason  of  the  deflection  is  as  follows  : 

A  wire  carrying  electricity,  becomes  practically 
itself  a  magnet ;  that  is,  a  straight  current  produces 
in  a  wire,  a  magnetic  field.  This  any  one  may  easily 
prove  for  himself,  by  passing  an  electrical  current 
through  a  wire  of  iron,  copper,  brass,  or  any  other 
metal,  and  permitting  the  wire  to  dip  into  a  heap  of 


31 

iron  filings.  The  filings  will  instantly  cling  to  the 
wire,  and  all  round  it,  just  as  if  it  was  a  natural 
magnet.  The  electric  wire  having  thus  virtually 
been  transformed  into  a  magnet,  when  placed  beside 
the  magnetic  needle,  interferes  with  the  attraction 
of  the  earth,  and  pulls  the  magnetic  needle  to  one 
side. 

The  case  is  simply  a  very  weak  but  very  near 
magnet,  i.  e.,  the  current-carrying  wire  acting  on  a 
poised  magnetic  needle  in  opposition  to  a  vexy 
strong  but  very  distant  magnetic  pole — the  north 
pole  of  the  earth  ;  and  thus  the  needle,  being  acted 
upon  by  both  oppositely,  takes  up  a  halfway  posi- 
tion, as  it  were,  "  on  the  fence."  The  earth's  mag- 
netism tends  to  make  the  needle  point  north  and 
south  ;  the  electric  current  acting  on  the  needle 
tends  to  make  it  assume  a  position  pointing  east 
and  west.  The  resultant  force  will  of  course  be 
between  the  two,  and  will  depend  on  their  relative 
strength.  If  the  current  is  very  strong,  the  needle 
will  turn  a  long  way  round,  but  never  farther  than 
to  a  complete  right  angle. 

But  we  have  so  far,  only  considered  the  effect  of 
one  parallel  wire.  If  we  want  a  greater  deflection, 
and  our  battery  power  cannot  conveniently  be  in- 
creased, what  is  to  be  done  ? 

If  we  cannot  increase  the  battery,  we  can  increase 
its  power  of  acting  upon  the  needle,  by  using  a 


32 

parallel  wire  on  both  sides  of  the  needle  ;  for  if  we 
carry  our  conducting  wire,  first  over  the  needle  from 
north  to  south,  and  then  back  from  south  to  north 
under  the  needle,  the  effect  will  be  doubled.  If  the 
wire,  instead  of  making  only  one  such  convolution 
round  the  needle,  were  to  make  two,  the  force 
would  again  be  doubled  ;  and  if  several  convolutions 
are  wound  round  the  needle,  the  force  would  be 
increased  nearly  in  proportion  to  the  number  of 
convolutions ;  we  say  nearly,  because  the  current  is 
itself  weakened  by  the  additional  length  of  wire 
required. 

If  the  convolutions  are  greatly  multiplied,  so  as 
to  form  a  coil,  the  force  is  enormously  increased,  and 
we  have  what  the  first  constructor  of  the  galvano- 
meter called  a  "multiplier." 

All  galvanometers,  therefore,  consist  of  a  coil  of 
insulated  wire,  and  a  magnetic  needle  delicately  sus- 
pended in  such  a  position  as  to  be  easily  deflected 
by  the  passage  of  a  current  of  electricity  through 
the  coil.  These,  with  the  addition  of  a  dial  plate, 
graduated  so  that  the  movements  of  the  needle  may 
be  interpreted,  are  the  only  absolutely  essential 
features  of  the  instrument. 

The  galvanometer  was  one  of  the  earliest  results 
of  Oersted's  discovery  ;  it  was,  indeed,  in  the  same 
year,  1820,  that  the  first  galvanometer  was  invented 
by  Professor  Johann  S.  C.  Schweigger,  of  Halle, 


who  passed  a  number  of  turns   of  insulated  wire 
round  a  compass  needle,  thus  multiplying  the  effect 
of  the  electricity,  and  constructing  a  galvanometer. 
An  instrument  of  this  rudimentary  character  is 
shown  in  Figure  9. 


Figure  9. 

Although  the  ordinary  galvanometer  constructed 
as  above,  is  very  well  adapted  to  detect  the  presence 
or  indicate  the  direction  of  a  current,  and  for  some 
simple  measurements,  especially  for  those  in  which 
the  deflection  is  not  greater  than  fifteen  or  twenty 
degrees,  it  is  not  to  be  depended  upon  for  any  test- 
ing in  which  a  greater  deflection  is  produced,  for 
the  following  reason  :  that  when  a  needle  is  de- 
flected, it  is  not  in  the  same  position  in  its  coil  as 
when  at  zero  ;  the  greater  the  deflection,  the  farther 
is  the  needle  moved  away  from  the  position  where 
its  coil  most  powerfully  influences  it,  and  the  nearer 
the  needle  approaches  the  right  angle,  at  which 
point  the  coil  has  no  influence  on  it  at  all,  the 
weaker  does  the  action  of  the  current  become.  In 


34 

order  to  overcome  this  difficulty,  and  for  other 
mathematical  reasons,  galvanometers  have  been  in- 
vented in  which  the  tangent,  or  sine,  of  the  angle 
of  deflection  is  proportional  to  the  strength  of  cur- 
rent measured.  These  are  called  tangent,  or  sine 
galvanometers. 

THE    TANGENT    GALVANOMETER. 

The  tangent  galvanometer  consists,  broadly  speak- 
ing, of  a  ring  having  a  groove  on  its  edge  filled  with 
insulated  wire,  and  provided  with  a  needle  which 
must  not  be  longer  than  one  sixth  of  the  diameter 
of  the  ring,  hung  or  pivoted  precisely  in  its  center, 
as  shown  in  Figure  10. 


Figure  10. 

One  of  the  best  tangent  galvanometers  is  that 
known  as  the  "  Western  Union  Standard,"  as  made 
by  J.  H.  Bunnell  &  Co.,  of  New  York. 

This  instrument  is  mounted  on  a  circular  hard- 
rubber  base,  7^  inches  diameter,  provided  with 


35 

. 

levelling  screws  and  anchoring  points.  The  Gal- 
vanometer consists  of  a  magnetized  needle  %  inches 
in  length,  suspended  at  the  center  of  a  rubber  ring 
6  inches  in  diameter,  containing  the  coils.  The 
coils  are  five  in  number,  of  the  resistances,  o,  i,  10, 
50,  and  150  ohms.  The  first  is  a  stout  copper  band 
of  inappreciable  resistance  ;  the  others  are  of  dif- 
ferent sized  copper  wires  carefully  insulated.  Five 
terminals  are  provided,  the  plug  holes  of  which  are 
marked  respectively  o,  i,  10,  50,  and  200. 

The  ends  of  the  coils  are  so  arranged  that  the 
plug  inserted  at  the  terminal  marked  200,  puts 
in  circuit  all  the  coils ;  at  the  terminal  marked 
50,  all  except  the  150  ohm  coil;  and  so  on,  till 
at  the  zero  terminal  only  the  copper  band  is  in 
circuit 

Fixed  to  the  needle,  which  is  balanced  on  jewel 
and  point,  is  an  aluminum  pointer  at  right  angles, 
extending  across  a  5-inch  dial  immediately  beneath. 
On  one  side,  the  dial  is  divided  into  degrees ;  on 
the  other  it  is  graduated,  the  figures  of  the  scale 
corresponding  to  the  tangent  of  the  angles  of  de- 
flection. 

The  needle  is  sometimes  suspended  by  a  fibre  of 
silk,  and,  of  course,  can  be  readily  provided  with 
coils  wound  to  any  required  resistance. 

The  instrument  is  made  complete  and  convenient 
by  being  provided  with  a  mahogany  case,  and  a 


strap  for  carrying  it. 
Figure  1 1. 


36 


This  instrument  is  shown  in 


Figure  n. 

When  this  instrument  is  used  to  measure  cur- 
rents, the  strength  of  the  current  is  proportional  to 
the  tangent  of  the  angle  of  deflection. 

For  the  benefit  of  the  non-mathematical  experi- 
menter, we  may  explain  that  a  tangent  is  a  line 
drawn  at  right  angles  to  one  of  the  diameters  of  any 
circle,  and  touching  the  circumference,  as  in  Figure 
12,  A  D  is  a  tangent  to  the  circle  B  C  E  F. 


37- 


Figure  12. 

In  the  case  of  the  tangent  galvanometer  the  dial 
of  the  instrument  is  the  given  circle,  and  the  zero 
point  is  the  point  at  which  the  tangent  touches  the 
circle.  The  tangent  is,  therefore,  an  imaginary  line, 
which  must  be  parallel  to  that  diameter  which  con- 
nects the  degree  of  ninety  on  one  side  to  the  same 
degree  on  the  other  side,  and  at  right  angles  to  the 
diameter  or  line  connecting  the  two  zero  points. 
Let  us  suppose  that  the  circle  is  the  dial  of  a  gal- 
vanometer marked  off  into  degrees,  and  that  the 
needle,  by  a  given  current,  is  deflected  to  twenty- 
seven  degrees  ;  double  the  current  strength  will  not 
double  the  deflection,  making  fifty-four  degrees, 
but  will  produce  a  deflection  which,  carried  out, 


38 

will  show  double  the  distance  measured  on  the 
tangent  scale,  and  that  deflection  will  be  foity-five 
degrees. 

In  mathematical  tables,  the  tangent  of  forty-five 
degrees  is  i,  therefore  that  of  twenty-seven  degrees 
is  .5,  or  thereabouts;  sixty-four  degrees  shows  a 
tangent  of  2,  seventy -two  degrees  3,  and  seventy-six 
degrees  4  ;  all  the  intermediate  degrees,  of  course, 
producing  proportional  fractions  on  the  tangent 
scale. 

Thus  we  see  that  a  current  producing  a  deflection 
of  seventy-six  degrees  on  a  tangent  galvanometer 
is  just  four  times  as  strong  as  a  current  producing 
a  deflection  of  forty-five  degrees  on  the  same  gal- 
vanometer. If  a  tangent  galvanometer  is  graduated 
to  degrees  only,  when  it  is  used,  to  obtain  correct 
results,  we  must  reduce  the  degrees  to  tangents  by 
means  of  a  table  of  tangents.  A  tangent  and  sine 
table  will  be  found  at  the  end  of  this  book  for 
reference  in  such  cases. 

Some  of  the  best  tangent  galvanometers  now 
made — for  example,  the  one  we  have  described 
— have  their  dials  graduated  in  addition  to  the 
degrees,  with  a  scale  corresponding  to  the  tangent 
divisions. 

THE    SINE    GALVANOMETER, 

Is  one  in  which  the  coils  are  made  moveable,  so 


39  - 

as  to  be  capable  of  revolving  on  the  axis,  around 
which  the  needle  turns. 

The  needle  is  pivoted,  or  suspended,  horizontally. 
A  scale  graduated  with  degrees  is  attached  to  the 
coil  and  a  pointer  fixed  on  the  base,  so  that  the 
angle  through  which  the  coil  is  turned  can  be 
observed. 

.  When  the  needle  is  deflected  by  a  current  pass- 
ing through  the  coil,  the  coils  are  turned  by  hand, 
following  the  needle  in  its  deflection  ;  as  the  coils 
are  thus  turned,  they,  of  course,  maintain  their 
power  on  the  needle,  and  it  accordingly  diverges 
still  more,  but  the  angle  it  makes  with  the  coils 
becomes  less  and  less,  until  at  length  a  point  is 
attained  at  which  the  needle  remains  parallel  with 
the  coil. 

When  this  point  is  reached,  the  influence  of  the 
earth's  magnetism  exactly  balances  the  deflective 
force  of  the  current.  The  strength  of  the  current 
that  produces  the  deflection  will  then  be  directly 
proportional  to  the  sine  of  the  angle  through  which 
the  coil  is  turned. 

As  shown  in  Figure  13,  the  sine  of  any  number 
of  degrees  is  that  part  of  the  diameter  of  a  circle 
which  is  included  between  a  line  drawn  from  the 
center  to  the  zero  point  of  the  graduation  circle, 
and  another  line  parallel  to  the  first,  cutting  the 
circle  at  the  degree  whose  sine  is  required. 


40 


Thus,  in  the  figure,  A  B  C  is  a  semi-circle  which 
we  may  suppose  to  be  the  graduated  scale  of  a  sine 
galvanometer.  B  D  is  the  line  from  the  center  to 
the  zero  point,  and  X  Z  and  E  F  are  parallel  lines 
cutting  the  circle  respectively  at  ten  and  twenty 
degrees,  the  space  on  the  diameter  A  C  between  Z 
and  D  being  the  sine  of  ten  degrees,  and  the  space 
between  F  and  D  the  sine  of  twenty  degrees. 

The  sine  of  ninety  degrees,  or  a  right  angle,  is  in 
sine  tables  called  1000,  that  of  one  degree  17,  and 
the  sines  of  all  degrees  between  i  and  ninety,  will 
be  found  in  the  table  at  the  end  of  this  book.  - 

If  a  current  of  known  strength,  then,  deflects  the 
needle  to  an  angle  of  thirty  degrees,  and  the  current 
to  be  compared  deflects  the  needle  to  angle  of  forty- 
five  degrees,  the  strength  of  the  second  current  is 
to  the  first  as  the  sine  of  forty-five  degrees  is  to  the 
sine  of  thirty  degrees. 

It  is  customary,   as   in  the   use   of  the  tangent 


41. 

galvanometer,  to  read  off  the  degree,  and  refer  to  a 
table  of  sines  for  the  required  sine. 

The  sine  galvanometer  is  not  so  convenient  for 
general  use  as  the  tangent  galvanometer,  and  is  con- 
sequently but  little  used,  except  for  scientific  ex- 
periments and  for  measuring  and  comparing  weak 
currents. 

THOMSON'S  REFLECTING  GALVANOMETER, 

Is  the  most  sensitive  galvanometer  known,  and 
is  very  useful  in  measuring  very  high  resistances, 
such  as  the  insulation  of  cables. 

Its  moving  parts  are  very  light  and  small,  the 
needle  being  a  little  piece  of  watch  spring  about  a 
quarter  of  an  inch  long.  This  is  suspended  by  a 
silk  fibre  in  the  center  of  a  coil  consisting  of  a  great 
number  of  turns  of  wire. 

Instead  of  fastening  a  pointer  to  the  needle,  as  in 
ordinary  galvanometers,  the  indications  are  made  by 
attaching  a  mirror  about  the  size  of  a  silver  five  cent 
piece  to  the  needle.  A  graduated  scale  is  placed 
two  or  three  feet  from  the  mirror,  and  a  beam  of 
light  derived  from  the  reflection  of  a  lamp  by  the 
mirror  shines  upon  the  scale,  which  is  usually  grad- 
uated to  360  divisions  on  either  side  of  the  zero 
point,  which  is  in  the  center  of  the  scale. 

Whenever  the  needle  moves,  the  beam  of  light, 
of  course,  moves  with  it,  and  is  thus  equivalent  to 


42 

an  index  needle  of  great  length  and  of  no  weight,  and 
a  very  small  movement  of  the  needle  produces  a  con- 
siderable movement  of  the  spot  of  light  on  the  scale. 

The  coil  completely  surrounds  the  needle  so  that 
the  needle  is  always  under  its  influence  irrespective 
of  its  angle  of  deflection. 

This  galvanometer  is  often  made  astatic,  and  is 
extremely  accurate  as  well  as  extremely  sensitive. 
The  infinitesimal  current  developed  by  dipping  a 
brass  pin  and  a  steel  needle  into  a  drop  of.  salt 
water,  will,  when  the  needle  and  pin  are  connected 
by  wires  with  the  galvanometer  terminals,  send  the 
spot  of  light  swinging  across  the  scale. 

The  coils  are  sometimes  wound  with  German 
silver  wire,  in  order  that  the  resistance  may  be  little 
affected  by  changes  of  temperature  ;  the  increased 
resistance  of  German  silver  over  copper  being  of  no 
consequence  when  the  instrument  is  used  to  measure 
high  resistances. 

The  resistance  of  a  Thomson  galvanometer  so 
wound  is  sometimes  as  high  as  50,000  ohms,  and 
such  an  instrument  with  but  one  cell  of  Daniell 
battery  would  give  a  large  deflection  through  ten 
million  ohms. 

ASTATIC    GALVANOMETERS. 

There  are  two  ways  of  increasing  the  sensitive- 
ness of  a  galvanometer.  The  first  method,  which 


43 

consists  in  increasing  the  effective  action  of  the  cur- 
rent by  coiling  a  great  many  convolutions  of  the 
insulated  conducting  wire  round  the  needle,  we 
have  already  considered.  The  second  method  is  to 
decrease  the  opposing  influence  of  the  earth's  mag- 
netism which  tends  to  keep  the  needle  pointing 
north  and  south. 

It  is  very  clear  that  unless  this  is  done,  the  cur- 
rent has  to  keep  up  a  constant  fight  against  the 
power  of  the  earth's  directive  action  on  the  needle. 
Therefore  it  is  also  clear  that  if  we  can  neutralize 
that  force  in  the  earth's  magnetism,  which  tends  to 
keep  the  needle  pointing  north  and  south,  the  cur- 
rent will  have  so  much  more  force  to  exert  on  the 
needle,  and  will  consequently  be  able  to  deflect  the 
needle  much  easier.  We  can  do  this  by  placing  two 
magnetized  needles  on  one  axis  (which  may  be  a 
light  wire  of  brass  or  aluminum),  with  their  poles 
placed  oppositely,  the  north  pole  of  one  over  the 
south  pole  of  the  other,  and  vice  versa,  as  in 
Figure  14. 


A 


Figure  14. 


44 

When  such  a  combination  is  used,  the  earth  will 
attract  the  upper  needle  and  tend  to  keep  its  N 
pole  pointing  northward  ;  but  it  will  repel  the  lower 
needle  with  an  equal  force,  and  tend  to  make  it 
turn  completely  round.  Thus  the  magnets  so  ar- 
ranged have  very  little  tendency  to  place  themselves 
north  and  south,  because  the  force  acting  on  one  is 
directly  counterbalanced  by  the  force  acting  on  the 
other. 

If  the  two  needles  were  exactly  equal  in  power, 
they  would  have  no  tendency  to  point  in  any  par- 
ticular direction,  for  any  directive  action  in  one  will 
be  counteracted  by  an  exactly  equal  and  opposite 
action  in  the  other.  In  practice,  one  needle  is  made 
a  little  stronger  than  the  other,  so  that  the  pair  has 
sufficient  tendency  to  point  north  and  south  to 
enable  it  to  regain  its  position  after  having  been 
deflected. 

On  the  same  axis  with  the  needles  we  may  place 
a  pointer,  A  B,  to  denote  the  deflections. 

The  nearer  the  two  needles  are  to  one  another, 
in  magnetic  strength,  the  slower  will  be  the  vibra- 
tions of  the  needle,  and  the  greater  the  delicacy  of 
the  galvanometer. 

A  needle  constructed  in  the  manner  described 
above  is  called  an  astatic  needle.  When  an  astatic 
needle  is  placed  in  a  coil  so  that,  as  shown  in  Figure 
14,  the  lower  needle  is  within  the  coil  and  the  upper 


45 

one  above  it,  its  deflections  will  be  much  greater 
than  if  an  ordinary  needle  were  used,  for  two 
reasons  :  in  the  first  place,  the  power  which  keeps 
the  needle  in  its  fixed  position  is  small,  and  the 
needle  is  consequently  more  easily  influenced ;  in 
the  second  place,  the  force  of  the  coil  is  exerted  in 
the  same  direction  on  two  needles  instead  of  one, 
for  the  upper  needle  being  much  nearer  the  upper 
part  of  the  coil  than  the  lower,  is  deflected  by  it 
alone,  and  the  deflection  so  induced  is  in  the  same 
direction  as  that  of  the  lower  needle. 

An  astatic  needle  so  mounted  in  a  coil  constitutes 
an  astatic  galvanometer. 

THE  DIFFERENTIAL  GALVANOMETER 

Is  one  which  has  a  needle  poised  or  suspended 
like  that  of  the  tangent  or  sine  galvanometers,  but, 
unlike  them,  the  needle  is  acted  upon  by  two  coils 
of  equal  length  and  resistance,  insulated  from  one 
another  with  great  care.  These  coils  each  surround 
the  needle  with  an  equal  number  of  convolutions, 
which  in  each  wire  are  equidistant  from  it. 

When  this  galvanometer  is  used,  one  end  of  each 
coil  is  connected  with  one  pole  of  the  battery,  and 
the  other  end  of  each  coil  with  a  wire  leading  to  the 
other  pole  of  the  battery  in  such  a  way  that  the 
current  flows  in  opposite  directions  through  the  two 
wires.  Now,  if  the  current  in  both  coils  is  of  the 


46 

same  strength,  one  tends  to  deflect  the  needle  to  the 
right  and  the  other  to  the  left,  and  the  needle  being 
pulled  with  equal  force  in  both  directions,  remains 
at  rest. 

If,  now,  one  current  be  made  stronger  than  the 
other,  the  balance  will  be  destroyed,  and  the  needle 
can  be  moved  by  the  stronger  current. 

If  -we  wish  to  measure  an  unknown  resistance 
with  this  galvanometer,  we  insert  the  resistance  to 
be  measured  in  the  circuit  of  one  of  the  coils.  This, 
of  course,  weakens  the  current  in  that  coil,  and  con- 
sequently its  effect  on  the  needle,  which  no  longer 
remains  balanced,  but  deflects  to  one  side.  If  we 
now  insert  a  rheostat  in  the  other  side,  and  unplug 
resistance  until  the  needle  again  balances  or  comes 
to  zero,  we  know  that  the  current  in  each  coil  must 
again  be  equal,  and,  therefore,  that  the  unknown 
resistance  in  the  circuit  of  one  coil  must  be  exactly 
equal  to  the  resistance  unplugged  from  the  rheostat. 

SHUNTS. 

Although  shunts  are  rarely  necessary  in  ordinary 
measurements,  at  least,  when  the  tangent  galvano- 
meter or  Wheatstone  Bridge  systems  are  employed, 
any  treatise  on  the  galvanometer  would  be  so  mani- 
festly incomplete  if  they  were  not  noticed,  that  we 
regard  it  as  eminently  proper  to  devote  a  short 
space  to  them  and  their  uses,  the  more  especially  as 


47 

experimenters,  and  other  persons  making  frequent 
measurements,  may  occasionally  desire  to  prepare 
shunts  for  themselves. 

When  the  differential  galvanometer  is  used,  the 
shunt  is  almost  an  essential,  and  it  is  also  useful  in 
such  direct  measurements  where  the  deflection  of  the 
needle  would  be  so  great  as  to  be  untrustworthy. 

A  shunt  may  be  defined  as  a  contrivance  for 
leading  by  another  route  part  of  the  current  which, 
as  a  whole,  is  too  powerful  for  the  immediate  pur- 
pose. In  the  present  connection,  it  is  a  coil  of 
wire  used  to  divert  some  definite  proportion  of 
a  current  aside  from,  or  past  the  galvanometer, 
instead  of  allowing  it  to  pass  through  the  galvano- 
meter coils.  For  example,  if  the  galvanometer  has 
its  two  terminals  united  by  a  wire  having  a  resist- 
ance equalling  one  ninety-ninth  of  the  galvanometer 
resistance,  we  reduce  the  galvanometer  to  one 
hundredth  of  its  original  sensibility,  ninety-nine 
hundredths  of  the  current  passing  through  the  shunt 
and  the  remaining  100  through  the  galvanometer. 

Similarly,  if  the  shunt  be  exactly  equal  to  the 
galvanometer,  the  current  will  divide  in  equal  pro- 
portions between  the  galvanometer  and  the  shunt. 
If  the  shunt  is  one  half  the  resistance  of  the  gal- 
vanometer, two  thirds  of  the  current  will  pass 
through  the  shunt  and  one  third  through  the 
galvanometer,  and  so  on. 


48 

The  rule  is,  that  the  current  divides  between  the 
galvanometer  and  the  shunt,  in  inverse  proportion 
to  their  respective  resistances,  the  greater  portion 
of  the  current  always  going  through  the  smaller 
resistance,  and  the  smaller  portion  through  the 
greater  resistance. 

Galvanometers  requiring  shunts  are  usually  pro- 
vided with  three,  which  are  respectively  one  ninth, 
one  ninety-ninth,  and  one  999th. 

These  reduce  the  amount  of  current  passing 
through  the  galvanometer,  respectively  to  its  one 
tenth,  one  hundredth,  or  one  thousandth  part. 

The  formula  for  finding  what  the  resistance  of  a 
shunt  should  be  to  give  it  a  definite  value,  is  to 
make  the  resistance  of  the  shunt  equal  to  that  of 
the  galvanometer,  divided  by  the  multiplying  power 
required,  minus  one.  For  example  :  Suppose  we 
have  a  galvanometer  whose  resistance  is  100  ohms, 
and  we  wish  to  prepare  a  shunt  which  will  reduce 
the  sensitiveness  to  one  tenth  ;  we  divide  the  gal- 
vanometer resistance  by  the  fractional  part  to  which 
we  desire  to  reduce  the  sensibility,  minus  one ;  that 
is,  we  divide  the  100  by  10  minus  i,  which  is  of 
course  9.  The  quotient  of  100  ohms  divided  by  9 
is  1 1  ohms  and  one  ninth,  which  is  the  resistance  of 
the  required  shunt,  and  is  one  ninth  of  the  resistance 
of  the  galvanometer.  This  is  called  a  shunt,  having 
a  multiplying  power  of  10. 


49 

To  obtain  the  true  value  of  a  deflection  taken 
from  a  shunted  tangent  galvanometer,  we  have  to 
multiply  the  tangent  by  the  multiplying  power  of 
the  shunt  used.  To  ascertain  the  multiplying  power 
of  any  shunt  whose  resistance  is  known,  we  divide 
the  resistance  of  the  galvanometer  by  the  resistance 
of  the  shunt,  and  add  one  to  the  quotient.  For 
example  :  we  are  using  a  galvanometer  with  a  re- 
sistance of  100  ohms,  and  we  insert  a  shunt  whose 
resistance  we  know  to  be  25  ohms ;  to  find  out  by 
what  number  we  have  to  multiply  the  shunted 
result,  we  divide  the  100  by  25,  which  gives  us  a 
quotient  of  4,  to  which  must  be  added  i,  showing 
that  5  is  the  multiplying  power  required. 

It  is  proper  to  state  that  when  we  use  a  shunt, 
the  current  which  passes  through  the  galvanometer 
is  not  strictly  the  proportionate  part  of  the  original 
current  to  which  it  is  apparently  reduced,  because 
by  the  act  of  employing  the  shunt  we  furnish  a 
double  route  for  the  current,  and  thereby  diminish 
the  external  resistance  of  the  circuit,  and,  as  a  con- 
sequence, the  strength  of  current  furnished  by 
the  battery  is  increased.  It  is,  therefore,  the  in- 
creased current  instead  of  the  original  one  that 
splits  between  the  shunt  and  the  galvanometer. 

To  illustrate  :  If  we  are  using  a  tangent  galvano- 
meter, and  the  tangent  of  deflection  without  the 
shunt  is  .80,  we  would  naturally  have  supposed  that 


50 

on  the  introduction  of  a- shunt  which  reduces  the 
sensitiveness  of  the  galvanometer  one  half,  the 
tangent  would  also  be  brought  down  one  half,  that 
is  to  .40.  But  such  is  not  the  case,  the  result  being 
some  higher  tangent  than  .40  ;  and  to  bring  about  an 
accurate  result,  we  must  first  find  the  joint  resistance 
of  the  shunt  and  galvanometer,  and  then  ins,ert  an 
additional  resistance  in  the  battery  circuit  equal  to 
the  amount  by  which  the  original  resistance  was 
decreased.  Thus,  if  both  the  galvanometer  and  shunt 
are  100  ohms  resistance,  the  joint  resistance  of  the 
two  is  50  ohms. 

In  this  case,  therefore,  we  should  have  to  insert 
50  ohms  in  the  battery  circuit,  to  compensate  for 
the  decrease  in  resistance,  and  to  bring  the  current 
back  to  its  original  strength. 

RHEOSTATS    AND    RESISTANCE    COILS. 

The  name  Rheostat  was  originally  given  by 
Wheatstone,  to  an  instrument  which  he  devised 
for  the  purpose  of  varying  at  will  the  amount  of 
resistance  in  a  circuit. 

It  consisted  of  two  cylinders  of  equal  diameter, 
one  of  brass  and  the  other  made  of  some  non- 
conducting material.  These  were  fixed  near  each 
other,  and  a  fine  German  silver  wire  was  wound  in 
opposite  directions  round  the  cylinders,  its  ends 


51 

being  in  electrical  connection  with  the  metallic  axes 
of  the  cylinders. 

The  axes  of  the  cylinders  were  connected  with 
two  binding  screws  by  means  of  sliding  contacts. 
The  part  of  the  wire  which  does  not  lie  on  the 
metal  cylinder  is  the  only  part  that  produces  resist- 
ance between  the  binding  screws,  and  by  winding 
the  wire  from  one  cylinder  to  the  other,  resistance 
could  be  added  to  or  taken  from  the  circuit.  This 
apparatus  is  scarcely  ever  now  used,  but  its  name 
survives,  and  is  now  often  applied  to  a  set  of  stan- 
dard resistance  coils,  arranged  together  in  a  box, 
and  adapted  for  use  in  electrical  measurements. 

What  we  now  call  a  rheostat,  is  a  box  containing 
a  number  of  spools  filled  with  insulated  wire,  the 
resistance  of  the  wire  on  each  spool  being  equal  to 
some  multiple  or  sub-multiple  of  the  ohm,  the  unit 
of  resistance.  The  several  coils  are  arranged  as  in 
the  figure. 


4: 


52 

The  cover  of  the  box  is  generally  of  hard  rubber, 
and  a  series  of  connecting  pieces  of  brass,  a,  b,  c,  are 
placed  on  it.  The  ends  of  the  several  brass  pieces 
are  very  close  together,  but  do  not  touch,  and  holes 
are  bored  between  the  ends  for  the  insertion  of 
brass  plugs  with  hard  rubber  handles,  so  that  when 
all  the  plugs  are  inserted,  there  is  an  unbroken  line 
of  conducting  brass  plates  all  along  the  cover  of 
the  box. 

Binding  screws  are  fixed  to  the  ends,  so  that  any 
desired  connections  can  be  made.  Each  of  the  coils 
in  the  box  below  has  its  terminals  united  to  two  of 
the  plates,  as  in  the  figure,  coil  i  is  attached  to  the 
plates  a  and  b,  coil  2  to  b  and  c,  so  that  every  coil 
joins  one  of  the  brass  plates  to  the  next. 

The  different  coils  may  be  of  any  required  resist- 
ance, and  may  be  varied  indefinitely.  They  are 
usually  made  to  increase  consecutively,  as  for  exam- 
ple, i,  2,  5,  10,  20,  50,  100,  500  ohms,  and  so  on. 

If  all  the  plugs  are  in  their  places,  there  is  prac- 
tically no  resistance  between  the  terminal  binding 
screws,  because  if  the  rheostat  is  placed  in  circuit, 
the  current  would  have  the  short  route  along  the 
brass  plates  on  the  cover  of  the  box. 

But  if  any  of  the  plugs  be  taken  out,  the  coil  of 
that  section  is  brought  into  the  circuit,  and  the  cur- 
rent can  pass  from  one  brass  plate  to  the  next  where 
the  plug  is  withdrawn  only  by  traversing  the  coil 


53 

» 

thus  introduced  into  its  path,  and  just  so  much 
resistance  as  is  represented  by  that  coil,  is  added  to 
the  entire  resistance  of  the  circuit. 

Thus,  if  the  coils  shown  in  the  figure  have 
respectively  5,  10,  25  and  100  ohms,  and  the  plug- 
between  b  and  c  be  withdrawn,  25  ohms  resistance 
is  thereby  thrown  into  the  circuit  between  the 
binding  screws. 

We  see,  then,  that  by  withdrawing  any  or  all  of 
the  plugs,  we  can  introduce  less  or  more  resistance. 

Numbers  representing  the  various  resistances  of 
the  coils  are  usually  placed  opposite  the  holes,  and 
by  adding  together  the  numbers  unplugged,  we 
ascertain  the  total  resistance  inserted. 

The  wire  used  in  resistance  coils  is  generally  made 
of  German  silver,  because  the  resistance  of  that 
alloy  changes  very  little  with  variation  of  tempera- 
ture ;  it  is  insulated  with  silk,  and  always  wound 
double,  as  shown,  so  as  to  neutralize  any  inductive 
action  of  the  convolutions  on  each  other,  and  also 
to  prevent  the  coils  from  affecting  galvanometers 
near  them  ;  when  so  arranged,  the  current  flows  at 
the  same  time  in  two  opposite  directions  round  the 
spool,  effectually  preventing  any  inductive  troubles. 
After  the  spools  are  wound  and  correctly  propor- 
tioned, they  are  saturated  with  hot  paraffine,  by 
which  their  insulation  is  maintained,  and  dampness 
prevented. 


54 

Thick  wire  is  generally  used  for  the  small  resist- 
ances, and  fine  wire  for  the  higher  ones. 

It  is  usual  to  so  arrange  the  different  resistances, 
that  by  properly  combining  them,  any  value,  from  a 
fraction  of  an  ohm  to  10,000  ohms,  can  be  obtained. 


Figure  16. 


The  resistance  box  can  be  of  any  suitable  shape, 
and  in  the  figure  a  circular  box  of  coils  is  shown. 
When  so  made,  it  is  very  convenient  to  pack  in  a 
box  with  the  galvonometer,  for  conveyance  from 
place  to  place. 


56 
THE    WHE ASTON E    BRIDGE. 

The  Wheatstone  Bridge,  or  balance,  though 
usually  classed  with,  and  explained  in  connection 
with  galvanometers,  is,  strictly  speaking,  not  a  gal- 
vanometer, but  a  system  of  measurement ;  or  an 
arrangement  of  circuits  whereby  a  galvanometer 
may  be  most  advantageously  employed. 

It  will  doubtless  be  a  surprise  to  some,  to  hear 
that  Wheatstone's  bridge  was  not  the  invention  of 
Wheatstone.  Such  however  is  the  case  ;  it  was 
invented  by  Mr.  S.  Hunter  Christie,  who  described 
it  before  the  Royal  Society  of  London,  in  1833,  in 
which  year  he  also  published  a  paper  regarding  it,  in 
the  Philosophical  Transactions.  He  called  it  a 
"  differential  arrangement "  and  used  it  for  measur- 
ing the  relative  conduction  of  different  metals,  and 
also  as  a  means  for  discovering  the  conducting  power 
of  metals  at  different  temperatures. 

Notwithstanding  all  Mr.  Christie's  efforts  to  make 
his  invention  known,  it  remained  in  oblivion,  until 
again  brought  forward  by  Sir  Charles  Wheatstone, 
who  in  1843,  ten  years  later,  wrote  an  important 
paper  on  electrical  measurement,  stating  the  source 
from  which  the  instrument  was  derived,  and  giving 
Christie's  dates  of  invention,  and  publication. 

It  was,  therefore,  from  no  fault  of  Wheatstone, 
that  his  name  has  become  inseparately  coupled  with 
that  of  the  bridge,  since  he  did  his  best  to  ascribe 


56 

due  credit  therefore  to  Christie,  who  unfortunately 
was  ten  years  too  early  to  be  appreciated. 

Even  Wheatstone,  however,  made  no  attempt  to 
use  variable  resistances  in  the  two  arms,  he  always 
using  equal  resistances,  and  consequently  having  his 
limit  of  measurement  greatly  circumscribed. 

Werner  Siemens,  in  1847,  was  the  ^rst  to  con" 
struct  a  bridge  with  arms  provided  with  variable 
resistances,  thereby  largely  amplifying  the  range  of 
the  instrument. 

We  will  here  describe  the  construction  of  the 
bridge,  and  afterwards  attempt  to  elucidate  the 
principle  of  its  operation. 

Figure  1 7  shows  the  theoretical  arrangement  of  a 


W 


w 


57  ' 

Bridge  system,  consisting  of  a  lozenge-shaped,  four- 
sided  figure,  two  of  its  corners,  i  and  2,  being  united 
by  the  wires  W  to  the  two  poles  of  a  battery,  and 
the  two  other  corners,  3  and  4,  connected  by  a 
cross  wire  and  galvanometer  G. 

This  was  the  original  form  of  the  balance  ;  but 
in  practice  it  is  rarely  now  so  constructed,  but  is 
arranged  usually  in  a  more  suitable  form  for  actual 
work. 

Yet  it  is  very  convenient  in  describing  the  bridge 
to  use  the  lozenge  form,  because  it  is  a  figure  readily 
borne  in  mind,  and  with  it,  the  principles  involved 
are  easier  of  explanation. 

Referring  to  the  diagram,  we  see  that  a  current 
of  electricity  starting  from  the  positive  pole  of  the 
battery,  on  arriving  at  the  point  i,  has  two  paths 
before  it ;  one  by  C,  4,  D  and  2,  the  other  by  A,  3, 
B  and  2,  both  routes  re-uniting  at  the  point  2,  and 
thence  proceeding  by  wire  W  to  the  negative  pole  of 
the  battery,  thus  completing  the  circuit. 

These  parallel  routes  are  called  the  branches, 
arms,  or  sides  of  the  bridge. 

The  current  therefore  divides  at  i,  and  if  the 
resistances  of  each  path  are  equal,  as  we  may  for  the 
present  assume  they  are,  half  of  the  current  goes  by 
A  B,  and  the  other  half  by  C  D,  to  the  point  2,  and 
from  that  point,  the  two  currents  go  together  back 
to  the  battery.  Now,  whatever  the  resistances  of 


58 

the  lines  A  B  and  C  D  may  be,  so  long  as  they  are 
equal  to  one  another;  that  is,  so  long  as  A  B  is 
equal  to  C  D,  the  galvanometer  G  will  not  be 
affected  in  any  way ;  its  needle  will  not  be  deflected, 
although  it  is,  as  we  see,  connected  with  both  lines 
by  a  cross  wire. 

Let  us  suppose  that  A  and  C  each  have  a  resist- 
ance of  10,  and  B  and  D  each  a  resistance  of  30 
ohms ;  which  will  of  course  be  equivalent  to  a  total 
resistance  of  40  ohms  on  each  side  ;  there  will  in  this 
case  be  no  deflection  of  the  galvanometer  needle, 
because  A  B  is  equal  to  C  D,  both  having  the  same 
resistance. 

Furthermore,  when  A  bears  the  same  proportion 
to  B,  that  C  does  to  D  ;  or  when  A  bears  the  same 
proportion  to  C,  that  B  does  to  D  ;  or  when  A 
multiplied  by  D  is  equal  to  C  multiplied  by  B,  no 
current  will  pass  through  the  cross  wire,  and  there 
will  be  no  deflection  of  the  needle. 

If  we  suppose  A  to  have  a  resistance  of  10,  B  of 
200,  C  of  100,  and  D  of  2000  ohms,  the  above  con- 
ditions are  fulfilled  ;  for  10  is  to  200  as  100  is  to 
2000 ;  and  10  bears  the  same  proportion  to  100,  as 
200  does  to  2000;  moreover  10  multiplied  by  2000 
is  equal  to  the  product  of  200  multiplied  by  100. 

With  these  resistances,  therefore,  the  proper  pro- 
portion exists,  balance  is  established,  and  no  deflec- 
tion of  the  needle  is  produced. 


59 


We  thus  see  from  the  above,  that  if  we  know  three 
of  the  resistances,  we  may  easily  find  the  fourth ; 
and  that  it  is  by  the  proportion  subsisting  between 
the  resistances  of  the  arms  of  the  bridge,  that  the 
resistance  of  the  fourth  can  be  calculated,  when  the 
resistances  of  the  other  three  are  known. 

If  we  have  known  resistances  at  A  and  C,  a  rheo- 
stat or  adjustable  resistance  box  in  the  branch  B, 
and  the  object  to  be  measured,  or  unknown  resist- 
ance in  D,  and  then  vary  the  resistance  in  B,  until 
the  needle  comes  to  zero,  we  may  be  sure  that  the 
unknown  resistance  in  D  bears  the  same  propori  ion 
to  that  unplugged  in  the  rheostat  B,  that  the  resist- 
ance of  C  does  to  that  in  A. 

This  of  course  is  readily  calculated  by  simple  pro- 
portion or  the  "rule  of  three." 

We  will  presently  see  why  this  should  be  so.  It 
has  already  been  stated,  (page  1 1)  that  an  electrical 
current,  or  a  steady  and  constant  transfer  of  electricity 
between  any  two  points,  is  caused  by  a  difference  of 
potential  between  such  points.  We  have  seen  that  if 
we  connect  the  two  poles  of  a  battery  by  a  long  wire, 
such  as  a  telegraph  line,  the  potential  will  fall  regu- 
larly and  gradually  from  the  end  which  is  united  to  the 
copper  or  positive  pole  of  the  battery  ;  and  will  rise 
regularly  and  gradually  from  the  end  which  is  attach- 
ed to  the  zinc  or  negative  pole,  until  they  both  meet 
in  the  middle  of  the  line,  where  the  potential  is  zero. 


60 

We  have  also  stated  that  if  instead  of  connecting 
the  poles  of  the  battery  by  a  metallic  conductor,  we 
put  one  pole,  say,  the  negative  to  a  ground  wire,  and 
attach  the  other  to  a  linewire  or  other  resistance 
connected  with  the  earth  at  a  distant  point,  so  that 
a  steady  current  flows  through  the  wire  to  earth,  the 
potential  will  fall  regularly  through  the  resistance 
of  the  line,  from  its  highest  figure  at  the  junction  of 
the  line  with  the  positive  battery  pole,  to  zero  at  the 
distant  ground  end. 

It  is  very  evident  from  the  above  that  if  a  current 
is,  as  we  have  said,  caused  by  a  difference  of  potential 
between  two  points,  there  can  be  no  current  between 
two  points  which  are  of  the  same  potential,  because 
there  is  no  force  tending  to  produce  one.  Just  as 
we  may,  after  stating  that  steam  is  produced  by 
heating  water  to  a  certain  temperature,  go  on  to 
show  that  if  the  water  is  not  heated,  steam  cannot 
be  produced. 

To  apply  the  idea  to  the  Wheatstone  Bridge,  let 
us  consider  the  construction  and  arrangement  of  the 
bridge  circuits. 

A  conducting  wire  attached  to  the  positive  pole  of 
a  battery  E  (Fig.  1 7),  is  extended  to  a  point  i,  where 
it  splits  into  two  conducting  wires,  A  B  and  C  D. 

These  wires  traverse  different  routes,  and  re-unite 
at  a  point  2,  and  from  the  point  2  a  wire  proceeds 
to  the  negative  pole  of  the  battery  E. 


61 

It  is  obvious  that  the  peculiar  form  of  the  arrange- 
ment is  not  essential :  the  main  point  being  that  the 
circuit  must  always  be  split  at  some  point  into  two 
branches,  which  at  some  other  point  are  again  united. 

Now  the  potential  must  certainly  at  the  point  i, 
where  the  circuit  splits,  be  the  same  for  both  wires ; 
and  as  certainly  at  the  point  2,  where  the  wires  re- 
unite, the  potential  must  also  be  the  same  for  both 
wires  ;  and  if  the  resistances  of  the  two  branches 
A  B  and  C  D  are  equal,  the  fall  of  potential  will  go 
on  just  as  if  the  two  wires  were  one,  and  will  there- 
fore be  equal  at  any  two  points  equally  distant  in 
resistance  from  .the  point  i;  consequently,  whatever 
the  potential  is  at  the  point  3  in  A  B,  it  likewise  is 
at  the  point  4  in  C  D,  and  when  we  unite  those 
points  by  a  cross  wire  and  galvanometer,  no  current 
passes  between  them,  and  no  deflection  of  the  needle 
can  occur. 

Nor  could  there  be  any  current  in  a  bridge  wire 
connecting  the  points  A  and  C,  or  B  and  D,  because 
their  potentials  also  are  equal.  But  if  we  connect 
A  and  D,  or  C  and  B,  by  a  cross  wire  or  bridge,  the 
needle  of  the  galvanometer  in  the  bridge  will  deflect 
violently,  because  A  is  much  nearer  than  D  to  the 
point  i,  as  also  C  is  much  nearer  than  B,  and  the 
difference  of  potential  naturally  causes  a  current 
between  the  two  points,  flowing  through  the  galva- 
nometer. 


62 

Let  us  see  now,  what  will  happen,  if  the  resist- 
ances of  the  branches  A  B  and  C  D  are  unequal.  If 
A  bears  the  same  proportion  to  C,  that  B  does  to 
D,  no  current  will  pass  between  3  and  4 ;  for  still, 
the  potential  is  the  same  for  both  branches,  at  the 
terminal  points  i  and  2,  and  no  matter  what  the 
resistances  are,  between  those  points,  if  any  two 
points,  such  as  3  and  4,  stand  at  the  same  propor- 
tion of  their  respective  resistances  from  the  point 
i,  the  potential  of  those  points  must  be  the  same, 
and,  therefore,  no  current  can  flow  in  a  cross  wire 
connecting  them. 

To  illustrate  :  let  us  suppose  that  the  point  i,  con- 
nected with  the  positive  pole  of  the  battery  has  a 
potential  of  20  plus,  and  the  point  2,  connected  with 
the  other  pole,  a  potential  of  20  minus,  making  a 
total  difference  of  40  between  the  two  points.  In 
this  case  we  assume  the  two  branches  to  be  equal, 
each  having  a  resistance  of  100  ohms.  Let  3  and  4 
be  points  exactly  in  the  center  of  each  resistance,  so 
that  the  resistance  between  3  and  A  is  50  ohms,  and 
the  resistance  between  3  and  B  is  also  50  ohms,  the 
same  being  true  of  4  and  C,  and  4  and  D.  The 
potential  of  the  points  3  and  4  is,  of  course,  a  figure 
just  halfway  between  that  of  i  and  that  of  2,  viz  : 
zero ;  and  no  current  can  pass  through  the  galvan- 
ometer, because  there  is  no  difference  of  potential 
between  the  points  3  and  4. 


63 

Let  A  and  C  be  exactly  halfway  in  resistance  be- 
tween 3  and  i,  and  4  and  i;  that  is,  the  resistances 
between  i  and  A,  and  between  i  and  C,  are  each  25 
ohms ;  the  potential  at  those  points  must  be  of 
course  just  one  fourth  of  the  total  difference,  i.  e.y 
10  plus  ;  again,  both  points  being  equal,  there  can  be 
no  current  in  a  cross  wire  connecting  them. 

Now  let  B  and  D  be  exactly  halfway  in  resistance 
between  3  and  2  and  4  and  2,  (their  respective  po- 
tentials thus  being  10  minus,)  and  connect  A  and  D 
by  a  cross  wire  and  galvanometer :  these  potentials 
are  not  equal,  that  of  A  being  10  plus,  and  that  of  D 
10  minus,  a  total  difference  of  20;  therefore  a  current 
flows  and  the  needle  deflects.  If  the  branches  are 
not  equal,  for  example,  if  A  has  a  resistance  of  10, 
B  of  100,  C  of  20,  and  D  of  200,  the  total  resistance 
in  A  B  thus  being  no,  and  in  C  D  220,  one  third 
of  the  current  will  pass  through  C  D,  and  two  thirds 
through  A  B.  Yet  the  difference  of  potential  is  the 
same  between  i  and  2,  and  must  necessarily  fall 
along  the  two  branches  exactly  as  before,  so  that  as 
there  is  between  i  and  3  a  resistance  of  10  ohms, 
being  one  eleventh  of  the  entire  resistance  A  B,  and 
between  i  and  4,  a  resistance  of  20  ohms,  that  is, 
one  eleventh  of  the  total  resistance  C  D,  the  poten- 
tial 20  plus  at  the  point  i,  must  at  the  points  3  and 
4  have  fallen  in  the  same  proportion,  one  eleventh 
of  the  difference  between  20  plus  and  20  minus,  and 


64 

the  potentials  at  3  and  4  are  equal ;  therfore,  no 
current  passes  between  those  points,  and  the  needle 
in  the  bridge  wire  does  not  deflect. 

The  principle  is,  we  see,  that  of  balancing  poten- 
tials against  one  another,  without  any  reference  to 
the  strength  of  current  passing,  and  when  we  try  to 
understand  it,  without  loading  it  down  with  two 
volumes  of  algebraic  equations  and  differential  calcu- 
lus, it  is  really  not  at  all  difficult  of  comprehension. 

COMPARISON    OF    DIFFERENT    SYSTEMS    OF 
MEASUREMENT. 

There  are  three  principal  methods  of  measuring 
resistances,  and  of  testing  lines.  These  are  :  First, 
by  the  angles  of  deflection  of  the  galvanometer 
needle.  Second,  by  matching  with  a  differential 
galvanometer  the  resistances  to  be  measured  with 
other  and  known  resistances,  until  the  needle  of  the 
galvanometer  shows  no  deflection.  Third,  by  the 
Wheatstone  Bridge,  or  Balance. 

We  may  use  the  first  general  method  in  two 
ways, — the  first  of  these  sub-methods  being  that  of 
substitution,  which  is  the  simplest  plan  in  use.  Any 
good  galvanometer  may  be  used  when  this  plan  is 
adopted. 

It  consists  in  placing  the  galvanometer  and  battery 
in  circuit  with  the  resistance  to  be  measured,  taking 


65 

note  of  the  amount  or  number  of  degrees  of  deflec- 
tion, and  then  replacing  the  unknown  resistance  by 
a  known  resistance,  for  example  a  rheostat,  and  ad- 
justing the  number  of  ohms  in  the  rheostat,  till  the 
deflection  of  the  needle  is  the  same  as  before. 

The  unknown  resistance  is  then  about  equal  to 
the  known  resistance  by  which  it  has  been  replaced. 

The  second  of  these  sub-methods  requires  a  sine 
or  tangent  galvanometer. 

It  consists  in  comparing  the  relative  strength  of 
current,  produced  by  the  same  battery  in  two  circuits, 
of  which  one  has  a  known  resistance,  while  the  other 
includes  the  resistance  to  be  measured. 

When  this  plan  is  followed  it  is  requisite,  either 
that  we  know  the  resistance  of  the  galvanometer 
and  battery  ;  or  that  those  resistances  must  be  so 
small  in  proportion  to  the  remainder  of  the  circuit, 
that  they  can  be  neglected  without  greatly  affecting 
the  result. 

In  using  this  method  we  may,  to  illustrate,  place 
a  tangent  galvanometer  having  a  resistance  of  100 
ohms,  a  battery  having  a  resistance  of  10  ohms,  and 
some  known  resistance,  say  100  ohms,  in  circuit 
together,  and  find  the  deflection  to  be,  let  us  say 
35  degrees.  Then  we  take  the  known  resistance, 
100  ohms,  out  of  circuit,  and  substitute  the  resist- 
ance to  be  measured,  finding  the  deflection  to  be 
altered,  say  to  25  degrees. 


66 

We  have  already  seen  that  the  strength  of  current 
passing  through  a  tangent  galvanometer  is  always 
proportional  to  the  tangent  of  the  angle  of  the 
needle's  deflection  ;  and  we  also  know  that  it  is 
inversely  proportional  to  the  resistance  in  circuit, 
that  is,  the  current  increases  as  the  resistance  de- 
creases, and  vice  versa. 

Therefore  we  refer  to  our  table  of  tangents,  find- 
ing the  tangent  of  35  degrees  to  be  .700  and  that  of 
25  degrees  to  be  .466  ;  the  strength  of  current  then 
may  be  calculated  by  direct  proportion,  tangent .  700 
being  to  the  strength  of  current,  when  the  known 
resistance,  100  ohms,  was  in  circuit,  as  tangent  .466 
is  to  the  current  strength  when  the  unknown  resist- 
ance was  substituted  therefore. 

The  unknown  resistance  is  calculated  by  inverse 
proportion,  tangent  .466  being  to  tangent  .700  as 
210  ohms,  the  total  original  resistance,  is  to  the 
total  resistance  in  circuit  when  the  unknown  resist- 
ance is  substituted  for  the  100  ohms,  viz  :  in  whole 
numbers  315  ohms. 

Deducting  from  this  the  resistance  of  the  galvano- 
meter and  battery,  i.  e.,  no  ohms,  the  unknown 
resistance  proves  to  be  205  ohms. 

For  several  reasons,  measurements  by  the  deflec- 
tion angles  cannot  be  depended  upon,  when  great 
accuracy  is  required. 

The  principal  reason   is  that  the  electromotive 


force  and  internal  resistance  of  the  battery  is  very 
apt  to  vary  during  the  time  of  making  the  measure- 
ments. It  is  also  sometimes  inconvenient  to  have 
to  'make  the  necessary  deductions  of  battery  and 
galvanometer  resistances,  particularly  when  we  do 
not  know  them. 

When  the  differential  galvanometer,  or  Wheat- 
stone  Bridge,  are  used,  such  defects  and  inconveni- 
ences are  obviated,  and  systems  of  measurement  in 
which  these  instruments  are  employed,  are  called 
"  null  "  methods,  because  the  results  are  obtained  by 
the  absence,  instead  of  by  the  amount  of  deflection  ; 
the  Wheatstone  Bridge  going  a  little  further  than 
the  differential,  and  giving  its  result,  not  only  by 
absence  of  deflection,  but  also  by  absence  of  current. 

It  is  clear  that  changes  in  the  battery  cannot 
affect  the  result  in  the  differential  galvanometer, 
because  as  the  current  goes  round  the  needle  simul- 
taneously, in  both  directions,  any  change  would  act 
in  both  coils,  and  being  thus  neutralized,  will  exer- 
cise no  effect  on  the  needle. 

If  the  differential  galvanometer  were  as  perfect 
practically,  as  it  is  theoretically,  it  would  be  a  most 
accurate  instrument,  but  in  practice  it  is  found  to 
be  very  difficult  to  adjust  the  two  halves  of  the  coil, 
so  as  to  have  exactly  equal  effects  on  the  needle  ; 
moreover,  imperfect  insulation  between  the  coils 
sometimes  occurs. 


68 

The  differential  galvanometer  has  therefore  been 
found  to  be  by  no  means  a  perfect  practical  instru- 
ment, and  it  has  consequently  been  in  a  great 
measure  superseded  by  the  Bridge,  which  has  none 
of  the  foregoing  defects,  and  which,  for  the  same 
bulk  and  cost,  has  far  greater  accuracy  and  sensi- 
bility within  wide  limits  than  the  differential  gal- 
vanometer. 

The  Wheatstone  Bridge  has  proved  to  be  in  every 
respect  the  most  successful  way  of  measuring  re- 
sistance. 

Its  advantages  lie  in  its  accuracy,  and  in  the  sim- 
plicity of  its  operation ;  and  in  the  fact  that  no 
special  form  of  galvanometer  is  required.  Any 
good  galvanometer  may  be  used  with  the  bridge  ; 
it  need  not  even  be  graduated ;  all  that  is  necessary 
is  that  it  shall  be  sensitive  enough  to  detect  the 
presence,  and  show  the  direction  of  a  current,  with- 
out in  any  way  determining  its  value,  or  comparing 
its  value  with  that  of  another  current. 

The  apparatus  of  the  Wheatstone  bridge  has  here- 
tofore usually  been  of  an  expensive  and  cumbrous 
character,  and  has  thus  been  beyond  the  reach  of 
the  rank  and  file  of  our  American  electricians,  oper- 
ators, and  inspectors,  who  are  not,  as  a  general 
thing,  rolling  in  wealth. 

An  erroneous  idea  has  also  to  a  great  extent  pre- 
vailed, owing  to  the  formidable  display  of  figures 


and  algebraic  symbols  indulged  in  by  text-books 
when  explaining  this  simple  apparatus  and  its  uses, 
that  both  the  instrument  itself,  and  the  methods  of 
employing  it,  were  complicated  and  difficult  in  the 
extreme.  These  reasons  have  militated  against  the 
extended  use  of  the  Bridge  in  this  country,  and  up 
to  the  present  time,  its  adoption  has  not  been  so 
universal  as  its  merits  would  seem  to  warrant. 

The  first  of  these  reasons  at  least,  no  longer 
exists,  since  complete  sets  of  bridge  apparatus  includ- 
ing the  galvanometer  and  all  necessary  resistances, 
and  having  sufficiently  wide  limits  of  measurement 
for  any  ordinary  work,  have  lately  been  introduced 
by  J.  H.  Bunnell  &  Co.,  of  New  York,  devised  and 
arranged  in  a  complete  form,  and  fitted  up  to  com- 
bine great  convenience  and  efficiency  in  work,  and 
excellency  of  finish,  with  prices  which  are  within 
the  reach  of  almost  everyone  ;  and  we  may  hope 
that  with  the  rapid  increase  of  knowledge  of  electri- 
cal science,  that  the  latter  reason  will  in  turn  soon 
cease  to  be. 


Description  of  the  Bunnell  Galvanometer,  Resistance 
Coils,  and  Wheatstone  Bridge. 


The  portable  bridge  apparatus  consists  of  a  gal- 
vanometer, and  a  full  set  of  resistances  arranged  as 


70 

a  bridge  system,  together  with  a  neat  morocco  case, 
in  which  both  instruments  may  be  placed  when  not 
in  use,  or  when  being  carried  from  place  to  place. 

The  galvanometer  is  made  thus :  A  delicate 
magnetic  needle  is  lightly  poised  in  pivots,  and 
carries,  at  right  angles  to  itself,  a  light  aluminum 
pointer,  extending,  on  both  sides,  over  a  dial  having 
a  scale  graduated  to  degrees ;  the  needle  is  sur- 
rounded by  three  coils  of  different  resistances.  The 
first  coil,  which  is  adapted  for  use  in  measuring 
high  resistances,  or  when  weak  currents  are  em- 
ployed, is  made  of  comparatively  fine  wire,  and  has 
a  resistance  of  70  ohms.  The  second  coil  which 
may  be  employed  with  medium  resistances  and  cur- 
rents, has  a  resistance  of  30  ohms,  and  the  third 
consists  simply  of  a  metal  band  which  passes  once 
or  twice  round  the  needle,  and  its  resistance  is  so 
small,  that  it  may  be  ignored. 

Each  coil  may  be  used  separately,  or  they  may 
be  used  in  series,  so  as  to  constitute  a  continuous 
coil  of  100  ohms,  as  will  be  hereafter  explained. 

The  instrument  is  handsomely  mounted  in  a  brass 
case  with  glass  cover,  and  is  set  on  a  base  of  vul- 
canite or  hard  rubber,  which  stands  on  three  adjust- 
able leveling  screws.  It  is  provided  with  a  damping 
device,  by  which  the  needle  is  maintained  in  a 
stationary  position  so  as  to  avoid  injury  when  not 
in  use,  or  while  being  carried  about :  and  has  also  a 


device  by  which  the  movements  of  the  needle  are 
ordinarily  limited  to  variations  of  a  very  few  degrees; 
this  arrangement,  by  means  of  a  projecting  handle, 
can  be  turned  round  the  case  so  as  to  be  useful, 
irrespective  of  the  extent  of  deflection. 

Figure  18  is  a  perspective  view,  and  Figure  19  a 
plan  view  of  the  galvanometer.      In  the  latter,  A  is 


Figure  18. 

the  rubber  base;  B,  B,  B,  the  leveling  screws;  i,  2, 
3  and  4  the  terminal  binding  screws,  C  the  coils ; 
N,  the  needle  pointer ;  L  the  damping  lever,  and  G 
the  handle  of  the  needle-guide  or  limiter. 

When  it  is  desired  to  use  the  entire  coil  of  100 
ohms^that  is  the   70  and  3oohm  coils  coupled  in 


72 


series,  one  of  the  connecting  wires  must  be  placed 
in  No.  i  binding  screw,  and  the  other  in  No.  4. 
When  the  3<>ohm  coil  is  to  be  used  alone,  the  wires 
are  placed  in  i  and  3. 


Figure  19. 

For  very  strong  currents,  and  very  low  resistances 
we  use  the  band  coil,  and  in  that  case  connect  the 
wires  in  i  and  2.  The  7o-ohm  coil  may  be  used  alone 
by  connecting  the  wires  with  terminals  3  and 


73 

This  instrument  is  sufficiently  sensitive  for  any 
ordinary  measurements,  and  with  one  cell  of  Le- 
clanche  battery,  gives,  through  a  resistance  of  12,000 
ohms,  using  the  loo-ohm  galvanometer  resistance,  a 
deflection  of  12  degrees;  the  7o-ohm  coil,  under 
similar  conditions,  giving  a  deflection  of  7,  and  the 
thirty-ohm  coil  a  deflection  of  6  degrees. 

With  no  external  resistance,  a  deflection  may 
also  be  produced  with  a  battery  consisting  of  a  steel 
sewing  needle,  and  a  piece  of  brass  immersed  in  a 
cup  of  water. 

THE    RESISTANCE    BOX,    OR    RHEOSTAT, 

Consists  of  a  stout,  but  handsomely-made  box  of 
metal,  having  a  mahogany  base,  and  a  hard  rubber 
cover,  on  which,  in  addition  to  the  plugs  for  con- 
trolling the  resistances,  there  are  six  binding  screws 
or  terminals,  two  of  which,  i  and  2,  are  for  the  bat- 
tery wires ;  two  others,  3  and  4,  for  the  attachment 
of  the  object  to  be  measured,  or,  as  in  the  case  of  a 
line  terminating  at  the  distant  end  in  a  ground  wire; 
for  the  line,  and  home  ground  wires ;  while  the  re- 
maining two,  5  and  6,  are  usually  for  the  galvan- 
ometer wires. 

Under  certain  conditions,  which  are  treated  of  in 
a  succeeding  page,  the  galvanometer  and  battery 
sometimes  change  places. 

Two  circuit  closing  keys  are  also  provided,  one  of 


74 

which,  K,  controls  the  battery,  and  the  other,  when 
depressed,  completes  the  galvanometer  circuit. 

A  lever  and  cam  is  fitted  to  the  battery  key,  by 
which  it  may  be  maintained  closed,  when  a  constant 
battery  current  is  required. 


Figure  20  is  a  theoretical  diagram  of  the  rheostat 
and  galvanometer,  when  connected  up  for  measure- 
ment ;  and  Figure  2 1  is  a  diagram  of  the  practical 
arrangement  of  the  rheostat  and  keys,  showing  the 
connections  of  the  galvanometer  and  battery. 


75 

By  comparing  the  theoretical  with  the  actual 
arrangements  in  the  figures,  it  will  be  at  once  seen 
that  the  electrical  connections  are  identical. 


In  the  holes  opposite  the  figures  are  placed  brass 
plugs  with  rubber  handles,  which,  when  withdrawn, 
introduce  resistance  in  ohms,  equal  in  amount  to 
the  figures  near  them. 


76 

Between  terminals  i  and  2  the  word  "  Battery  " 
is  printed,  between  3  and  4  the  word  "  Object,"  and 
between  5  and  6  the  word  "  Galvanometer ; "  the 
ordinary  method  of  connecting  the  instruments 
being  thereby  indicated,  so  that  no  mistake  need 
be  made  by  the  operator,  or  experimenter,  however 
inexperienced. 

The  resistances  in  the  two  branches  of  the  bridge 
are  10,  50,  100,  and  500  ohms  ;  and  in  the  com- 
parison coil  the  resistances  range  from  4,000  ohms 
to  one  hundredth  of  an  ohm  ;  the  total  resistance 
when  all  the  plugs  are  out  being  eleven  thousand 
one  hundred  and  eleven  ohms  and  one  tenth. 

In  the  figures,  the  corresponding  points  are  let- 
tered alike,  so  that  they  may  be  easily  compared ; 
the  branches  separate  at  the  point  C,  one  of  them, 
E,  leading  to  D,  where  it  unites  with  the  comparison 
coils  G  and  galvanometer  key  K2;  and  the  other,  F, 
to  the  binding  screw*  5,  where  it  unites  with  the 
\vire  leading  to  the  object  screw  4,  and  may  also  be 
connected  with  one  of  the  galvanometer  wires. 

The  comparison  coil  or  rheostat  G  leaves  the  first 
branch  at  D,  and  may  be  traced  through  the  several 
resistances  to  terminal  2,  where  it  unites  with  a  wire 
leading  to  the  object  screw  3,  and  where  it  is  also 
connected  with  the  return  battery  wire. 

In  measuring  resistance  with  this  instrument,  it 
will  generally  suffice  to  unplug  equal  amounts  of 


77 

resistance  in  the  branches  E  and  F  ;  then,  when  the 
galvanometer  needle  is  brought  to  zero,  all  we  have 
to  do  is  to  add  the  figures  of  the  resistances  un- 
plugged in  the  rheostat  which  forms  the  branch  G, 
the  total  being  equal  to  the  unknown  resistance 
required. 

This  method  may  be  utilized  to  measure  any  re- 
sistance between  1 1,100  ohms  and  one  hundredth  of 
an  ohm.  Therefore  it  is  plain  that,  even  if  the 
branches  E  and  F  were  unadjustable,  the  scope  of 
this  system  is  sufficiently  wide  for  the  majority  of 
measurements ;  but  as  E  can  be  made  larger  than  F, 
or  F  larger  than  E,  it  follows  that  the  limits  of 
measurement  can  be  greatly  enlarged.  For  if  we 
unplug  all  the  resistance,  660  ohms,  in  branch  E, 
and  but  10  ohms  in  branch  F,  we  may,  theoretically 
at  least,  by  unplugging  but  one  hundredth  of  an 
ohm  in  the  rheostat  G,  measure  as  small  a  resistance 
as  innnr  of  an  ohm,  for  as  660  is  to  10,  so  is  .01  of 
an  ohm  to  the  answer. 

Again,  by  unplugging  660  ohms  in  F,  10  ohms 
in  E,  and  unplugging  all  the  resistances  in  G,  we 
may  measure  a  resistance  as  great  as  733,332-rV 
ohms. 

Therefore,  by  these  methods  of  varying  the  branch 
resistances,  we  are  enabled  to  measure  resistances 
much  greater,  and  much  smaller,  than  those  in  the 
rheostat. 


78 


MANAGEMENT    OF    THE    GALVANOMETER    AND 
RHEOSTAT. 

In  using  the  rheostat  or  resistance  coils,  the  brass 
plugs  ought  to  be  always  clean  and  bright,  as  a  dirty 
or  oxidized  plug  does  not  completely  cut  its  coil  out. 

When  a  plug  is  placed  in  its  socket,  it  should  be 
slightly  twisted  so  as  to  produce  a  good  contact ; 
but  care  must  be  taken  not  to  twist  too  hard,  or  the 
hard  rubber  heads  may  be  twisted  off,  or  made  loose. 

It  is  always  in  order,  before  commencing  measure- 
ments, to  give  all  the  plugs  a  gentle  twist,  to  see  if 
they  are  tight ;  and  the  brass  part  of  the  plugs  should 
be  touched  as  little  as  possible.  In  using  the  rheo- 
stat as  a  bridge,  the  battery  key  should  always  be 
depressed  first,  and  the  galvanometer  key  afterwards, 
the  latter  being  depressed  only  long  enough  to  show 
the  direction  of  the  deflection,  until'  zero  is  nearly 
obtained,  when  it  may  be  firmly  pressed. 

These  precautions  tend  to  prevent  injury  to  the 
coil  and  needle  of  the  galvanometer,  from  sudden 
and  violent  currents. 

When  short  pieces  of  wire  or  low  resistances  are 
being  measured,  great  care  must  be  taken  not  to 
apply  the  battery  current  too  long,  as  the  coils 
may  in  that  case  become  heated,  and  their  resistances 
increased. 


One  of  the  first  things  to  be  done  in  using  a 
galvanometer  is  to  find  out  and  note  the  direction 
to  which  the  needle  deflects  under  the  influence  of  a 
given  current.  If  we  find  that  when  the  zinc  pole 
of  a  battery  is  connected  with  No.  i,  the  needle 
deflects  to  the  right,  we  may  know  that  by  reversing 
the  battery,  and  connecting  the  copper  or  carbon  pole 
to  that  terminal,  the  deflection  will  be  to  the  left. 

In  measurements  with  the  bridge,  it  is  well  to 
connect  the  battery  always  the  same  way ;  by  so 
doing  we  are  enabled  to  ascertain,  by  the  direction 
of  deflection,  whether  we  have  unplugged  too  much 
or  too  little  resistance  in  the  rheostat. 

Before  laying  the  galvanometer  aside,  the  needle 
should  be  made  stationary  by  the  damping  lever. 

Before  commencing  to  use  the  galvanometer,  it 
must  be  carefully  leveled  by  the  screws  provided  for 
that  purpose,  and  care  must  be  taken  that  the  needle 
swings  freely,,  and  points  directly  to  the  zero  mark. 

As  already  indicated,  the  three  coils  are  to  enable 
us  to  work  either  with  strong  or  weak  currents. 

It  is  a  good  plan  to  clean  the  points  of  both  bat- 
tery and  galvanometer  keys  occasionally. 

PRACTICAL    HINTS    FOR    USE    IN    WHEATSTONE    BRIDGE 
MEASUREMENT. 

As  the  branch  resistances  E  F  are  adjustable,  the 
testing  operator  has  the  option  of  choosing  any  of 


80 

them,  or,  if  he  pleases,  more  than  one.  The  ques- 
tion naturally  rises,  which  one,  or  what  is  the  best 
proportion  ?  It  may  be  demonstrated,  either  mathe- 
matically or  experimentally,  that  the  sensibility  or 
sensitiveness  of  the  instrument  is  greatest  when  the 
resistances  in  all  four  branches  are  alike. 

Now,  as  this  identity  can  only  occur  when  the 
resistance  to  be  measured  is  similar,  either  to  any 
one  of  the  resistances  in  E  and  F,  or  to  the  sum  of 
any  two  or  more  of  them,  it  is  obvious  that  the  next 
best  thing  is  to  make  the  resistances  in  the  branches 
E  F  as  near  to  the  resistance  we  are  going  to  meas- 
ure as  we  can. 

Of  course,  we  do  not  generally  know  what  the 
resistance  of  the  object  to  be  measured  is,  or  its 
measurement  would  be  a  superfluous  task,  but  we 
can  often  make  an  approximate  estimate  of  it,  and 
act  accordingly. 

For  instance,  if  we  have  a  resistance,  say,  the 
magnet  of  an  electric  bell,  and  suppose  it  to  be 
somewhere  in  the  neighborhood  of  15  ohms,  and 
we  desire  to  ascertain  the  actual  resistance  ;  we  first 
unplug  10  ohms  in  each  of  the  branches  E  and  F, 
because  10  is  the  branch  coil  nearest  in  magnitude 
to  our  estimate  of  15.  Again,  if  we  suppose  the 
unknown  resistance  to  be  nearer  50  than  it  is  to  10, 
we  withdraw  the  5o-ohm  plug  in  E  and  F ;  if  nearer 
100  than  50,  we  unplug  100  ohms  in  each  branch, 


81 

and  so  on.  If,  on  the  contrary,  we  can  form  no 
estimate  of  the  resistance  we  are  about  to  measure, 
we  can  get  it  approximately  by  a  rough  measure- 
ment, after  which  we  employ  the  right  arrangement 
as  above,  and  make  a  more  accurate  test. 

For  example,  we  wish  to  measure  a  line,  and  have 
no  idea  of  the  magnitude  of  its  resistance  :  we  un- 
plug 500  ohms  in  the  branches  E  and  F,  and  500 
also  in  the  rheostat ;  the  needle  deflects,  we  will  say, 
to  the  right ;  we  unplug  200  ohms  more  and  find 
that  the  needle  deflects  still  farther  to  the  right,  thus 
showing  that  we  are  going  in  the  wrong  direction  ; 
we  then  insert  the  plugs  again,  leaving  only  100 
ohms  unplugged  in  the  rheostat,  and  find  that  the 
needle  now  passes  the  zero  point  and  swings  a  little 
to  the  left,  showing  that  as  500  was  too  large,  100 
is  too  small.  At  this  point  we  have  ascertained 
that  the  unknown  resistance  is  considerably  less  than 
500  ohms,  and  is  greater  than  100  ohms.  We  may 
now  insert  the  5ooohm  plugs  in  the  branches,  and 
unplug  in  each  branch  100  ohms  in  place  thereof; 
the  needle  now  swings  a  little  more  to  the  left,  be- 
cause the  sensitiveness  is  increased ;  we  now  care- 
fully unplug  resistance  in  the  rheostat,  trying  the 
galvanometer  key  after  each  change,  and  find 
that  the  needle  comes  to  zero  when  we  have  with- 
drawn a  total  of  1 75  ohms,  which  is  the  resistance 
required. 


82 

For  the  great  majority  of  measurements,  the  re- 
sistance unplugged  in  E  should  be  equal  to  that  in 
F,  it  being  only  necessary  to  make  them  differ  when 
a  resistance  is  to  be  measured,  either  greater  than 
the  greatest  or  less  than  the  least  in  the  rheostat  G. 

It  may  be  thought  by  some,  that  to  measure  re- 
sistances smaller  than  10  ohms,  it  will  be  best  to 
have  no  resistance  unplugged  in  E  and  F. 

This  is  not  so  ;  it  is  absolutely  essential  that  there 
shall  be  some  resistance  in  E  and  F,  or  there  will  be 
no  deflection  of  the  needle,  the  galvanometer  in  that 
case  being  practically  short  circuited. 

It  will  be  always  found  economical  to  use  as  small 
a  battery  as  possible  ;  moreover,  the  liability  of  heat- 
ing and  so  altering  the  resistances  of  the  coils  is 
diminished  by  so  doing  ;  therefore,  for  all  low  resist- 
ances, one  or  two  cells  only  should  be  used,  and  in 
most  cases,  one  will  be  found  sufficient. 

Large  or  high  resistances,  of  course,  require  more, 
but  it  is  not  likely  that  more  than  ten  cells  at  the 
outside  will  ever  be  required. 

Schwendler  gives  the  following  rule  for  use  in 
Wheatstone  Bridge  systems :  For  all  resistances 
under  10,000  ohms,  one  Daniell  cell ;  above  10,000 
and  less  than  100,000  ohms,  ten  cells,  and  for  every- 
thing above  100,000  ohms,  one  hundred  cells. 

As  has  been  already  indicated,  the  Daniell,  or 
crowfoot,  gravity  battery  is  the  best,  on  account 


83. 

•» 

of  its  constancy  ;  but  the  Leclanche  answers  very 
well,  if  it  be  more  convenient  to  use  it. 

Ordinarily  the  galvanometer  is  placed  between  the 
points  D  and  5,  as  indicated  in  the  drawing,  and  as 
marked  on  the  Bunnell  rheostat. 

To  obtain  the  greatest  degree  of  sensitiveness 
with  a  given  cuirent,  the  rule  is,  however,  as  follows  : 

"  Of  the  two  resistances,  that  of  the  battery  and 
that  of  the  galvanometer,  connect  the  greater  resist- 
ance so  as  to  join  the  two  greatest  to  the  two  least 
of  the  four  other  resistances." 

To  connect  in  this  way,  we  should  generally  have 
to  connect  the  galvanometer  with  the  terminals 
marked  for  the  battery,  and  vice  versa,  which  is  un- 
desirable and  would  tend  to  confusion. 

Therefore  such  a  method  is  not  here  recommend- 
ed, and,  in  fact,  the  usual  method  of  connection 
will,  as  a  rule,  be  found  to  be  fully  as  sensitive  as  is 
necessary.  When  a  high  resistance  is  to  be  meas- 
ured, and  the  battery  power  at  hand  is  not  sufficient, 
it  is  a  good  plan  to  make  the  change,  and  see  if  im- 
proved results  ensue.  Such  cases  will  rarely  occur 
in  ordinary  practice,  and  the  matter  has  here  been 
referred  to,  more  for  the  sake  of  giving  as  full  in- 
formation upon  the  subject  as  possible,  than  for  any 
other  reason. 


84 


INSTRUCTIONS    AND    FORMULAE    FOR 
MEASUREMENT. 


In  using  the  resistance  coils  which  we  have  des- 
cribed, it  must  be  understood  that  the  branch 
resistances  E  F,  are  those  which  are  denoted  by 
full  lines  in  the  diagram,  Figure  21,  and  which 
stretch  along  the  sides  of  the  resistance  box  from 
the  binding  screws  on  either  side,  represented  by 
the  numbers  10,  50,  100  and  500  ohms;  while  the 
comparison  coils  or  rheostat  is  marked  in  dotted 
lines. 

In  the  succeeding  pages,  when  the  branches  are 
referred  to,  it  will  be  understood  that  the  resistances 
E  F  are  meant,  even  when  not  specifically  so  stated. 

In  like  manner  when  the  rheostat  is  spoken  of, 
without  a  letter  of  reference,  the  comparison  coil  G 
will  be  implied. 

Supposing  now,  that  we  are  about  to  measure 
resistances,  we  connect  the  instruments  as  follows  : 
Connect  the  rheostat,  battery  and  galvanometer  by 
wires,  as  in  the  diagram  Figure  22,  with  the  term- 
inal screws  of  the  rheostat-coil  facing  the  operator. 
The  object  to  be  measured,  whatever  it  may  be, 
must  be  attached  to  the  two  middle  binding  screws, 
which  are  marked  as  "  object"  screws,  and  which 
we  shall  hereafter,  for  brevity,  call  Nos.  3  and  4 ; 


85 

the  galvanometer  is  connected  with  the  two  left- 
hand  screws,  which  are  so  marked,  and  which  we 
may  call  5  and  6,  and  the  battery  must  be  connected 
with  the  two  right-hand  screws  T  and  2,  which  are 
also  marked  for  it. 

We  will  of  course  use  that  coil  of  the  galvan- 
ometer which  is  nearest  in  resistance  to  the  object 
we  are  about  to  measure.  It  is  not,  however,  ad- 
visable to  use  the  band  coil,  unless  the  resistance 
we  are  measuring  is  very  small  indeed. 


Figure  22. 

We  now  withdraw  the  proper  plugs  from  each  of 
the  branches  E  F,  and  unplug  resistance  in  the 
rheostat  G,  as  near  in  amount  as  we  can  guess,  to 
that  we  are  about  to  measure. 

Pressing  the  battery  key,  and  immediately  there- 
after the  galvanometer  key,  we  observe  which  side 
the  needle  deflects ;  we  unplug  a  few  ohms  more, 


86 

and  again  press  the  keys  ;  if  the  deflection  increases 
in  the  same  direction  as  before,  we  know  at,  once 
that  we  are  getting  farther  away  from  the  true 
resistance,  and  that  the  resistance  we  first  unplugged 
was  too  much.  If  on  the  contrary,  the  deflection 
is  lessened,  we  may  know  that  the  first  resistance 
was  not  sufficient,  and  we  may  then  fasten  the  bat- 
tery key  down,  and  change  or  remove  plugs  in  the 
rheostat  until  the  needle  comes  to  zero.  In  any 
case  where  the  battery  power  is  not  great,  the  bat- 
tery key  may  be  fastened  down  after  the  first  trial. 

If  the  deflection,  on  withdrawing  the  second  plug, 
passes  the  zero  point,  then  we  know  the  first  re- 
sistance was  too  little,  and  the  second  too  much. 
In  this  case,  also,  we  change  the  rheostat  plugs  un- 
til zero  is  obtained. 

When  the  needle  remains  at  zero,  and  does  not 
move  when  the  galvanometer  key  is  raised  and  de- 
pressed, balance  is  obtained,  and  the  measurement 
complete.  All  of  the  methods  here  given,  are  to 
be  used  with  the  Bridge,  except  when  otherwise 
stated. 

TO    MEASURE  A   RESISTANCE   NOT   GREATER  THAN  THE 

GREATEST,    OR    LESS    THAN    THE    LEAST    IN    THE 

RHEOSTAT. 

Let  us  now  assume  that  we  are  about  to  measure 
the  resistance  of  a  local  sounder.  We  see  that  it  is 


87' 

wound  with  rather  coarse  wire,  and  guess  it  to  be 
about  6  ohms.  We  connect  the  second  coil  of  the 
galvanometer,  and  unplug  10  ohms  in  each  of  the 
branches  E  F,  because  10  is  the  nearest  resistance 
to  6,  in  those  branches ;  we  also  unplug  6  ohms  in 
the  rheostat  G. 

Depressing  the  battery  and  galvanometer  keys, 
the  former  first,  we  find  the  needle  deflects  to  the 
right ;  replacing  the  i-ohm  plug  in  the  rheostat,  and 
thus  leaving  but  5  ohms  unplugged,  we  find  on  de- 
pressing the  keys  that  the  needle  now  goes  a  little 
to  the  left.  We  see  at  once  that  the  right  resistance 
is  not  so  much  as  6,  but  is  more  than  5  ohms.  We 
now  unplug  0.5,  or  half  an  ohm,  and  the  needle 
comes  to  zero.  All  we  have  now  to  do  is  to  note 
the  figures  unplugged  in  the  rheostat  and  we  have 
the  result  5.5  ohms,  which  is  the  resistance  of  the 
sounder. 

Let  the  object  be  a  relay,  having  an  estimated 
resistance  of  between  100  and  500  ohms.  In  this 
case  we  use  the  loo-ohm  galvanometer  coil,  and 
unplug  100  ohms  in  the  branches  E  F,  and  100 
also  in  the  rheostat.  We  fasten  down  the  battery 
key,  and,  as  before,  vary  or  remove  plugs  in  the 
rheostat  until  zero  is  obtained.  We  find  that  the 
needle  comes  to  zero,  when  we  have  unplugged  126 
ohms,  which  is  therefore  the  exact  resistance  of  the 
relay. 


88 

TO    MEASURE    A    RESISTANCE    SMALLER    THAN    THE 
SMALLEST    IN    THE    RHEOSTAT. 

The  resistance  of  a  certain  length,  say  three  feet, 
of  No.  1 8  copper  wire  is  required. 

Trying  to  measure  with  equal  resistances  in  E  F, 
the  resistance  is  found  to  be  smaller  than  .01  of  an 
ohm,  the  needle  refusing  to  come  to  zero. 

Unplug  50  in  the  branch  E,  and  10  in  branch  F, 
and  vary  the  rheostat  plugs  :  the  needle  becomes 
stationary,  for  example,  when  .07,  or  seven  hun- 
dredths  of  an  ohm,  are  unplugged  in  the  rheostat. 

Then  50  :  10  :  :  .07  :  .014  of  an  ohm,  that  is,  by 
simple  proportion  50  ohms  is  to  10,  as  seven  hun- 
hundredths  of  an  ohm  is  to  seven  five-hundredths, 
or  fourteen  thousandths,  of  an .  ohm,  which  is  the 
resistance  of  the  piece  of  wire. 

TO    MEASURE    A    RESISTANCE    LARGER    THAN    THE 
LARGEST    IN    THE    RHEOSTAT. 

The  exact  resistance  of  a  conductor,  having  a 
resistance  which  we  know  to  be  higher  than  any  in 
the  rheostat,  is  required.  Unplug  10  ohms  in  the 
branch  E,  and  500  in  F,  and  vary  the  rheostat  plugs 
as  in  other  cases  ;  suppose  zero  to  be  obtained  when 
a  resistance  4,500  ohms  is  unplugged. 

Then  10  is  to  500  as  4,500  is  to  the  required 
resistance,  i.  e.,  225,000  ohms. 


89 

Of  course  the  figures  in  all  of  the  above  are  given 
merely  for  the  sake  of  illustration.  Much  greater 
extremes,  both  higher  and  lower,  may  be  measured 
with  equal  facility  by  increasing  the  inequality  be- 
tween the  branches  E  and  F. 

To  measure  a  telegraph  line  in  metallic  or  loop 
circuit,  the  two  ends  of  the  circuit  are  connected  to 
the  resistance  box  terminals  3  and  4,  just  as  with 
any  other  conductor ;  and  the  procedure  is  then 
according  to  the  examples  given  above. 

TO    MEASURE    AN    ENTIRELY    UNKNOWN    RESISTANCE. 

It  has  been  stated  that  the  best  results  are  to  be 
obtained  with  the  bridge,  when  the  resistances  un- 
plugged in  the  branches  E  and  F,  are  those  nearest 
to  that  which  is  to  be  measured  ;  because  the  nearer 
the  four  sides  of  the  parallelogram  are  to  one  another 
in  resistance,  the  more  sensitive  is  the  galvanometer. 

If  we  have  no  idea  of  the  magnitude  of  the  resist- 
ance we  are  going  to  measure,  it  is  clear  that  we 
cannot  do  this.  In  such  a  case  Kempe  recommends 
the  following  method :  "Take  out,  say  the  loo-ohm 
plugs  in  E  and  F,  and  then,  having  adjusted  the 
resistance  in  the  rheostat  G,  so  as  almost  to  obtain 
equilibrium,  change  the  loo-ohm  plugs  in  E  F  for 
the  50  or  5oo-ohm  plugs,  and  see  if  the  deflection  is 
increased.  As  soon  as  the  plugs  which  produce  the 


90 

greatest  deflection  are  found,  we  can  then  finally 
vary  the  plugs  in  the  rheostat  until  exact  equilibrium 
is  attained.  Furthermore  the  same  device  may  be 
adopted  when  measuring  with  unequal  resistances 
in  E  and  F." 

TO    MEASURE    THE    RESISTANCE    OF    A    LOOP    OR 
METALLIC    CIRCUIT. 

Connect  the  two  ends  of  the  circuit  to  the  resist- 
ance box  binding  screws  3  and  4,  and  proceed  as 
before. 

TO   MEASURE   A   RESISTANCE    BY   SUBSTITUTION,    USING 

ANY   GALVANOMETER,    AND   HAVING   BOTH   TERMINALS 

OF    THE    RESISTANCE    AT    HAND. 

Connect  the  galvanometer  in  direct  circuit  with  a 
sufficient  battery  to  give  a  deflection  of  any  number 
of  degrees  not  over  40  or  under  15,  and  with  the 
object  to  be  measured,  and  note  the  deflection 
obtained. 

Remove  the  object  and  replace  it  by  a  rheostat ; 
unplug  resistance  in  the  rheostat,  until  the  same 
deflection  is  again  obtained,  when  the  resistance 
unplugged  will  be  equal  to  that  of  the  unknown 
resistance  measured  ;  that  is,  of  course,  subject  to 
any  change  which  may  have  taken  place  in  the 
condition  of  the  battery  ad  interim. 


91 

The  galvanometer  supplied  with  the  Wheatstone 
Bridge  set  is  not  suited  for  this  class  of  measurement, 
as  the  movement  of  -its  needle  is  limited  ;  it  being 
intended  solely  for  methods  of  measurement  in  which 
the  absence  of  a  deflection,  or  zero,  denotes  that  bal- 
ance is  obtained  between  a  known  and  unknown  re- 
sistance. It  is  also  too  sensitive  for  small  resistances. 

Any  tangent  or  sine  galvanometer  will  do,  or 
even  a  good  compass  galvanometer,  if  no  other  can 
be  obtained  ;  and  in  all  cases,  as  in  the  bridge  system, 
that  coil  of  the  galvanometer  (if  it  has  more  than 
one)  should  be  used,  which  approximates  the  most 
nearly  to  the  resistance  to  be  measured. 

To  diminish  the  time  consumed  in  substituting 
the  unknown  for  the  known  resistance  as  much  as 
possible,  a  button  switch  may  be  used,  as  shown  in 
the  diagram,  Figure  23. 

The  rheostat  provided  for  the  bridge  set  may  be 
used,  if  the  resistance  to  be  measured  is  not  greater 
than  the  total  resistance,  which  may  be  unplugged 
in  its  resistance  coils  ;  and  if  it  is  used,  the  connec- 
tions must  be  as  follows  :  Connect  one  pole  of 
the  battery  to  rheostat  post  No.  i,  and  the  other  to 
one  of  the  galvanometer  terminals ;  connect  the 
opposite  galvanometer  terminal  to  the  switch-but- 
ton, as  shown  in  the  figure. 

Attach  the  two  wires  of  the  unknown  resistance 
to  binding  screws  5  and  6  of  the  rheostat,  and  one 


92 

of  the  switch-studs  also  to  No.  6.  Finally,  unite 
post  2  of  the  rheostat  with  the  other  stud  of  the 
switch. 

In  measuring,  first  turn  the  switch  to  the  side 
that  connects  with  the  object  to  be  measured.  Ob- 
serve the  deflection,  call  it,  say,  30  degrees ;  turn 
switch  to  the  rheostat  side,  and  unplug  resistance 
until  the  needle  again  shows  30  degrees.  Turn  the 
switch  rather  quickly  once  or  twice  to  see  if  the 
same  deflection  is  maintained  on  either  side,  and 
when  no  change  is  observed,  add  up  the  resistance 
unplugged,  which  will  be  the  resistance  required. 

TO  MEASURE  BY  SUBSTITUTION  A  'RESISTANCE  OF 
WHICH  ONE  END  ONLY  IS  AT  HAND,  THE  OTHER 
BEING  CONNECTED  WITH  THE  EARTH  AT  A  DISTANT 

POINT. 

Suppose  the  unknown  resistance  to  be  a  tele- 
graph line,  and  our  apparatus  to  be  a  tangent  gal- 
vanometer, a  rheostat  and  a  battery :  connect  one 
pole  of  the  battery  to  earth,  and  the  other  with  one 
of  the  galvanometer  terminals ;  unite  the  other  gal- 
vanometer terminal  to  the  line  wire,  and  observe 
the  deflection. 

Now  disconnect  the  line  from  the  galvanometer, 
and  the  ground  wire  from  the  battery,  and  insert 
the  rheostat  in  their  place,  connecting  one  of  the 
rheostat  posts  with  the  battery,  and  the  other  with 


93 


the  galvanometer ;  unplug  resistance  in  the  rheostat 
till  the  galvanometer  needle  shows  the  same  deflec- 
tion as  when  the  line  was  in  circuit. 

The  unplugged  resistance  denotes  the  resistance 
of  the  line. 

The  button-switch  may  be  here  also  taken  ad- 
vantage of  to  save  time,  and  when  it  .is  used,  it 


k J 


Figure  23. 

must  be  so  connected  that  turned  one  way  the  bat- 
tery and  galvanometer  are  connected  with  the  line, 
and  when  turned  the  other  way  with  the  rheostat. 
If  we  have  no  other  rheostat  than  the  one  belong- 
ing to  the  bridge  set,  this  may  be  used  by  uniting 
one  battery  pole  with  post  i  of  the  rheostat;  the 
other  battery  pole  with  one  of  the  galvanometer 
terminals ;  post  2  of  the  rheostat  to  one  side  of  the 


94 

button-switch  ;  post  5  of  the  rheostat  to  a  ground 
wire,  and  post  6  of  the  rheostat  to  the  line  to  be 
measured,  and  also  to  the  other  side  of  the  button- 
switch.  The  button  or  movable  bar  of  the. switch 
is  then  attached  to  the  remaining  galvanometer  post. 

The  course  of  the  circuits  may  readily  be  traced 
out  by  reference  to  Figure  23. 

If  the  line  to  be  measured  is  long,  and  has  several 
relays  in  circuit,  the  result  will  be  the  resistance  of 
both  line  and  relays.  The  relays  ought  to  be  cut 
out  if  an  accurate  measurement  is  required.  The 
total  resistance  divided  by  the  -number  of  miles 
gives  the  resistance  per  mile. 

TO  MEASURE  BY  THE  WHEATSTONE  BRIDGE  THE  RE- 
SISTANCE OF  A  CIRCUIT  OF  WHICH  WE  ONLY  HAVE 

ONE  END. 

The  connections  in  this  case  are  shown  theoret- 
ically in  the  diagram  Figure  24,  and  are  practically 
made  by  uniting  the  line  wire  to  be  measured  to 
screw  4,  and  a  ground  wire  to  screw  3  of  the  rheostat. 
Connect  the  battery  poles  with  rheostat  screws  i  and 
2,  and  the  galvanometer  with  5  and  6,  as  indicated 
by  the  words  stamped  between  the  binding  screws. 

Now  proceed  according  to  general  instructions. 
Suppose  the  line  to  be  measured  to  be  30  miles 
long,  and  that  we  know  it  to  be  built  of  No.  12.  iron 
wire.  We  also  know  that  No.  12  galvanized  iron 


95 

wire  has  a  resistance  per  mile  of  about  32  ohms; 
this,  multiplied  by  30  for  the  number  of  miles,  gives 
an  estimated  resistance  of  960  ohms.  We  therefore 
unplug  from  the  branches  A  and  C,  a  reasonably 
high  resistance,  for  example,  500  ohms  in  each. 

We  then  vary  the  plugs  in  the  rheostat  B,  until 
the  needle  remains  at  zero.  Supposing  this  occurs 
when  980  ohms  are  unplugged,  980  ohms  is  the 
total  resistance  of  the  line,  being  a  little  over  the 
estimated  resistance. 


LINE 


Figure  24. 

TO    MEASURE    THE    INTERNAL    RESISTANCE    OF    A 
BATTERY. 

There  are  several  methods  of  determining  the  in- 
ternal resistance  of  a  battery ;  we  give  a  few  of  the 
easiest  and  best. 

ist — By  a  sine  or  tangent  galvanometer.  Con- 
nect the  galvanometer,  the  battery  to  be  measured, 


96 

and  a  rheostat  in  circuit  together,  and  observe  the 
deflection  ;  all  the  plugs  being  left  in  the  rheostat, 
the  only  resistance  in  circuit,  is  practically  that  of 
the  galvanometer  and  of  the  battery  itself.  If  the 
deflection  is  too  great,  unplug  any  convenient  resist- 
ance ;  once  more  observe  the  deflection  produced, 
and  find  its  tangent.  Halve  the  tangent,  and  by 
reference  to  the  table  find  what  deflection  the  halved 
tangent  corresponds  to. 

Now  unplug  resistance  in  the  rheostat,  till  the 
latter  deflection  is  produced.  The  total  resistance 
is  now  doubled. 

Deduct  the  resistance  of  the  galvanometer,  and 
twice  the  original  resistance  unplugged  in  the 
rheostat,  if  any,  from  the  amount  added  to  produce 
the  second  deflection,  and  the  remainder  will  be  the 
resistance  of  the  battery. 

For  example :  Suppose,  with  a  galvanometer 
having  a  resistance  of  100  ohms,  we  desire  to  meas- 
ure the  internal  resistance  of  a  certain  battery  ;  after 
joining  the  battery,  galvanometer,  and  rheostat  as 
described,  \ve  find  the  deflection  to  be  50  degrees ; 
we  consider  this  to  be  too  high  for  an  accurate 
measurement,  and  to  reduce  it,  we  unplug  60  ohms. 
This  brings  the  deflection  down  to  40  degrees. 

Referring  to  the  table  of  tangents,  we  find  the 
tangent  of  40  degrees  to  be. 839,  the  half  of  which 
is  .4195.  The  number  of  degrees  which  corresponds 


97 
• 

most  nearly  to  the  latter  tangent,  is  23  ;  we  there- 
fore unplug  enough  resistance  to  bring  the  deflection 
down  to  23  degrees,  let  us  say  300  ohms. 

Now  as  the  total  resistance  is  doubled,  it  follows 
that  the  current  is  halved.  To  ascertain  the  resist- 
ance of  the  battery,  we  double  the  resistance  first 
unplugged,  viz.:  60  ohms,  and  add  the  sum,  120 
ohms,  to  the  resistance  of  the  galvanometer,  100 
ohms  ;  then  deducting  the  total  220  ohms  from  the 
resistance  added  300  ohms,  we  find  the  remainder  to 
be  80  ohms,  which  is  the  internal  resistance  of  the 
battery.  Of  course,  if  we  do  not  find  it  necessary 
in  the  first  place  to  reduce  the  deflection  by  adding 
resistance,  it  is  only  necessary  to  subtract  the  re- 
sistance of  the  galvanometer  from  the  amount 
added  to  halve  the  tangent. 

2nd  Plan.  If  we  have  two  cells  exactly  alike,  we 
may  join  them  in  opposition  to  one  another,  so  that 
they  generate  no  current  of  their  own,  and  then 
measure  them,  either  with  a  tangent,  sine,  or  differ- 
ential galvanometer,  or  by  a  Wheatstone  Bridge, 
just  as  we  would  measure  any  other  resistance.  The 
resistance  of  one  cell  will  be  half  that  of  the  two. 

3rd  Plan.  This  method  is  shown  theoretically  by 
the  diagram,  Figure  25,  and  the  practical  arrange- 
ment in  connection  with  the  bridge  rheostat  will 
be  hereafter  described.  The  resistance  coils  R,  are 
joined  up  in  direct  circuit  with  the  galvanometer  G, 


and  the  battery  B,  which  is  to  be  measured,  and  a 
shunt  S,  is  placed  so  as  to  connect  the  poles  of  the 
battery  by  a  second  circuit,  parallel  to  that  in  which 
the  galvanometer  and  rheostat  are  included. 

The  shunt  should  have  a  resistance  exactly  equal 
to  the  other  external  resistances,  i.  e.,  the  galvano- 
meter and  whatever  resistance  is  unplugged  in  the 
rheostat,  and  these  again  should  be  proportionate 


mm — ^ 


s. 


Figure  25. 

to  the  supposed  resistance  of  the  battery  to  be 
measured.  When  connected  as  shown,  the  shunt, 
and  the  galvanometer  and  rheostat,  present  a  joint 
external  resistance  to  the  battery,  and  the  current 
from  the  battery  B,  divides  itself  through  the  two 
equal  resistances,  half  of  it  passing  through  the 
shunt  S,  and  the  other  half  through  the  galvan- 
ometer G  and  rheostat  R. 

Observe  the  deflection  of  the  needle  at  G.     Now 
remove  the  shunt ;  the  whole   current  must  then 


pass  through  G  and  R,  and  the  deflection  at  G  ac- 
cordingly increases.  The  external  resistance  be- 
tween the  poles  of  the  battery  is  now  double  what 
it  was  when  the  shunt  was  connected,  because,  by 
the  laws  of  parellel  or  derived  circuits,  the  resistance 
of  two  equal  circuits  together,  is  just  half  that  of 
either  of  them  alone  ;  but  the  internal  resistance  of 
B  remains  unchanged. 

Now  unplug  resistance  in  R,  till  the  same  deflec- 
tion is  produced  as  at  first,  and  the  extra  resistance 
unplugged  will  be  exactly  equal  to  the  internal  re- 
sistance of  the  battery  B. 

For  example  :  A  galvanometer  whose  resistance 
is  100  ohms,  and  a  battery,  the  resistance  of  which 
is  to  be  ascertained,  are  connected  in  circuit  with  a 
resistance  of  400  ohms  in  the  rheostat ;  a  shunt  of 
500  ohms  is  now  caused  to  connect  the  battery 
wires,  and  we  observe  the  deflection  to  be  24  de- 
grees. Removing  the  shunt,  the  whole  current 
passes  through  the  galvanometer,  and  the  deflection 
increases.  To  bring  it  back  to  24  degrees,  let  us 
assume  that  we  have  to  unplug  60  ohms,  which  is 
therefore  the  resistance  of  the  battery. 

To  use  the  Burmell  Bridge  rheostat,  the  connec- 
tions must  be  made  in  the  following  manner : 

The  poles  of  the  battery  to  the  rheostat  terminals 
i  and  6  ;  the  galvanometer  to  3  and  6  ;  also  by  a 
short  wire  5  and  6  are  to  be  joined.  To  remove 


100 

the  shunt,  take  out  the  short  wire  connecting  5 
and  6. 

4th.  The  fourth  plan,  and  in  some  respects  the 
best,  will  now  be  described.  It  is  often,  from  its 
discoverer,  called  Mance's  method,  and  it  consists  in 
placing  the  battery  whose  internal  resistance  is  to 
be  measured,  in  the  fourth  branch  of  a  Wheatstone 
Bridge,  and  varying  the  resistance  of  the  other 
branches. 


Figure  26. 


Referring  to  Figure  26,  which  clearly  shows  the 
arrangement,  B  is  the  rheostat,  A  and  C  the  other 
branches,  and  the  battery  is  inserted  at  D.  The 
galvanometer  is  kept  in  its  usual  place  on  the  cross 
wire,  and  a  key  K  is  put  in  the  usual  place  of  the 
battery,  by  which  we  can  close  or  open  the  circuit 
of  the  wire  1,2,  whenever  we  please. 


101 

» 

The  battery  when  connected  as  shown  in  the 
figure,  in  the  branch  D,  produces  a^  cte£tam  deflec* 
tion  of  the  needle;  the  resistance' ^ih'J  the 'other 
branches,  chiefly  that  in  the  rhebst^  >B,J  ^  ;£iou^ 
varied  and  adjusted,  not  until  the  needle  comes  to 
zero,  as  described  in  previous  measurements,  but 
until  the  deflection  of  the  needle  is  the  same, 
whether  the  key  is  pressed  or  not. 

When  this  condition  is  reached,  the  battery  re- 
sistance is  balanced  by  the  other  branches.  If  the 
resistance  unplugged  in  A  is  equal  to  that  in  C, 
the  amount  unplugged  in  B,  will  equal  the  required 
internal  resistance  of  the  battery  in  D.  If  A  and 
C  are  unequal,  the  required  resistance  will  as  usual 
be  found  by  proportion,  A  being  to  C  as  B  is  to  D. 

In  this  method  of  measuring  the  resistance  of 
the  battery,  the  galvanometer  resistance  need  not 
be  considered.  Another  advantage  is,  that  the 
electromotive  force  of  the  battery  need  only  be 
steady  during  the  short  periods  of  time  occupied  in 
depressing  and  raising  the  key. 

To  make  the  proper  connections,  we  connect  the 
battery  with  the  object  posts  3  and  4  ;  unite  posts 
i  and  2  together  by  a  short  wire,  and  connect  the 
galvanometer  as  usual  with  posts  5  and  6. 

In  measuring,  first  press  the  galvanometer  key, 
and  then  the  other.  This  method  as  described  in 
most  of  the  textbooks,  and  works  on  electrical 


102 

measurement,  is  apparently  very  easy  and  simple. 
Some  care,  however,  is  necessary  in  order  to  per- 
form it  successfully,  for,  with  a  galvanometer  of  any 
sensitiveness  at  all,  the  deflection  will  be  so  great  as 
to  be  unreliable,  and  sometimes  the  needle  will  even 
deflect  to  its  utmost  limits. 

This  is  a  point  on  which  most  of  the  textbooks 
are  dumb,  yet  it  is  a  most  important  one.  Some 
means  must  evidently  be  adopted  to  bring  the  de- 
flection within  reasonable  limits,  and  several  plans 
have  been  proposed.  This  may  sometimes  be  done 
by  making  the  branch  resistances  unequal,  balancing, 
for  "example,  100  in  A,  against  10  in  C,  or  even  a 
still  greater  inequality.  If  this  is  not  effectual,  the 
desired  end  may  be  gained  by  giving  the  needle  an 
initial  bias  to  one  side  by  means  of  a  permanent  bar 
magnet,  or,  what  is  equivalent  to  this,  bringing  the 
needle  nearly  back  to  zero  by  approaching  the  per- 
manent magnet  to  it. 

Or,  we  may  shunt  the  galvanometer  by  connecting 
its  terminals  by  a  cross  wire  of  suitable  resistance  ; 
an  adjustable  resistance  being  best,  of  course ;  if 
such  a  one  is  not  easily  attainable,  almost  any  may 
be  made  to  answer;  a  Morse  sounder,  for  example, 
may  be  looped  to  the  galvanometer  posts,  after  the 
other  connections  are  completed. 

Supposing  then,  that  the  galvanometer  has  a  re- 
sistance of  100  ohms,  and  the  sounder  5  ohms,  100 


103 

» 

parts  of  the  entire  current  will  pass  through  the 
sounder  coils,  and  5  parts  through  the  galvanometer. 
A  smaller  and  more  convenient  deflection  is  thus 
produced. 

Or  again,  we  may  insert  resistance  in  the  galvan- 
ometer circuit  between  it  and  the  rheostat  (that  is, 
at  any  point  in  the  bridge  wire),  until  the  deflection 
is  brought  low  enough. 

All  of  the  above  plans  are  easy  of  application,  and 
the  most  convenient  may  be  adopted.  The  writer  has 
very  successfully  employed  the  permanent  magnet. 

TO  ASCERTAIN  THE  RESISTANCE  OF  A  GALVANOMETER. 

There  are  several  methods  of  measuring  the 
resistance  of  a  galvanometer. 

If  we  have  a  second  galvanometer,  the  easiest  and 
most  obvious  way  of  ascertaining  the  resistance  of 
either,  is,  of  course,  to  regard  them  as  any  other  ordi- 
nary resistance  to  be  measured,  using  one  of  them 
as  an  instrument  with  which  to  measure  the  other, 
in  connection  with-  the  bridge,  as  already  described. 

But  we  often  find  it  desirable  to  know  the  re- 
sistance of  the  galvanometer  which  we  are  using, 
when  we  have  no  other  to  use  as  a  measuring 
instrument. 

We  will  describe  the  two  simplest  methods  of 
finding  the  galvanometer  resistance  under  such 
circumstances. 


104 

First.  Using  the  Wheatstone  Bridge.  This  plan 
is  the  counterpart  of  the  bridge  method  of  meas- 
uring the  internal  resistance  of  a  battery,  and  was 
derived  by  Sir  William  Thomson  from  that  method. 

Figure  27  is  a  diagram  of  the  arrangement,  A 
and  C  being  as  usual  the  branch  resistances,  and  B 
the  comparison  rheostat ;  the  galvanometer  is  placed 
in  branch  D,  instead  of  being  in  the  cross  wire  ;  in 


Figure  27. 

the  regular  place  of  the  galvanometer  is  a  circuit- 
closing  key,  so  that  we  may  easily  and  quickly 
connect  and  disconnect  the  two  points  which  would 
ordinarily  be  connected  with  the  galvanometer. 

The  battery  E  is  in  its  usual  position,  and,  of 
course,  the  current  flowing  from  it  passes  through 
the  branches  of  the  bridge,  causing  the  galvanometer 
needle  to  deflect. 


105 

The  resistances  in  the  other  branches,  principally 
the  branch  B,  are  then  adjusted  until  the  deflection 
remains  unaltered,  whether  the  key  in  the  cross 
wire  be  depressed  or  not.  When  this  occurs,  a 
balance  is  evidently  made,  and  consequently  we  get 
the  resistance  of  the  galvanometer  by  the  usual 
proportion  ;  thus,  as  A  is  to  C,  so  B  is  to  the  re- 
sistance of  the  galvanometer  in  D. 

If,  for  example,  we  have  100  ohms  unplugged  in 
A  and  C,  and  to  effect  a  balance,  we  have  to  unplug 
250  ohms,  the  first  two  branches  being  equal,  the 
galvanometer  in  D  is  also  equal  to  the  amount  un- 
plugged in  B,  that  is,  250  ohms. 

It  will  be  observed,  that  though  this  is  not  a  null 
method,  in  the  strict  sense  of  there  being  no  current 
in  the  galvanometer,  and  no  deflection  of  the  needle, 
it  is  so  in  view  of  the  fact  that  the  deflection  when 
balance  is  established,  does  not  change  when  a  cer- 
tain contact  is  made.  To  clearly  understand  this 
method,  we  must  refer  again  to  the  principle  of  the 
bridge.  From  what  has  already  been  explained  of 
the  operation  of  the  bridge,  it  is  easy  to  see  that 
when  we  are  measuring  an  ordinary  resistance  in 
the  usual  way,  before  the  balance  is  made,  a  current 
is  flowing  through  the  galvanometer,  or  its  needle 
would  not  be  deflected  ;  consequently,  if  we  make 
any  change  in  its  resistance,  the  strength  of  current 
flowing  in  all  the  branches  will  be  affected. 


106 

If,  on  the  contrary,  balance  is  established,  the 
needle  stands  at  zero,  therefore  no  current  is  passing 
through  the  galvanometer,  and  it  follows,  that  if  no 
current  is  passing  through  the  galvanometer,  that 
we  may  open  the  cross  wire,  or  even  take  away  the 
galvanometer,  or  make  any  other  change  in  the 
cross  wire  resistance,  without  affecting  the  currents 
in  the  branch  wires  at  all,  and  upon  these  principles 
the  above  method  of  measuring  the  resistance  of  a 
galvanometer  depends. 

We  connect  the  instruments  in  the  following 
manner: 

Battery  as  usual  to  the  rheostat  terminals  i  and  2. 
Galvanometer  to  the  object  terminals  3  and  4. 
Connect  also  the  ordinary  galvanometer  terminals 
5  and  6  by  a  short  wire. 

In  measuring,  we  first  depress  the  battery  key, 
then  galvanometer  key,  and  then  vary  the  plugs 
until  the  deflection  does  not  change  when  the  latter 
is  pressed.  When  we  get  a  deflection  that  only 
varies  a  little  when  we  press  the  galvanometer  key, 
we  may  fasten  the  battery  key  down,  and  manipulate 
the  galvanometer  key  only,  until  a  constant  deflec- 
tion is  reached.  The  same  caution  is  necessary 
in  using  this  method,  as  in  the  Mance  method 
of  measuring  the  resistance  of  a  battery,  and  for 
the  same  reason;  viz.:  the  extreme  deflection  of 
the  needle  which  ordinarily  will  take  place. 


107 

To  avoid  this,  the  first  thing  to  be  done,  is  to 
reduce  the  sensitiveness  of  the  galvanometer  by 
unplugging  unequal  proportions  in  the  bridge,  and 
making  the  galvanometer  side  of  much  the  highest 
resistance,  so  that  the  major  part  of  the  current  will 
pass  through  the  rheostat.  We  may,  for  example, 
unplug  10  ohms  in  A,  and  500  in  C,  and  then  vary 
the  resistances  in  B. 

We  may  also  reduce  the  battery  power  to  a  single 
cell.  If  the  current  is  still  too  strong  for  accurate 
measurements,  we  must  adopt  one  of  the  following 
expedients  : 

Either  shunt  the  galvanometer  by  a  resistance 
which  is  known,  and  measure  the  joint  resistance 
of  the  shunt  and  galvanometer  in  parallel  circuit ; 
and  afterwards  calculate  the  resistance  of  the  gal- 
vanometer, from  the  difference  between  the  resist- 
ance of  the  parallel  circuit,  and  the  known  resistance 
of  the  shunt. 

Or  shunt  the  battery  by  a  resistance  sufficient  to 
reduce  the  deflection  to  reasonable  limits. 

Or  weaken  the  current  by  inserting  "sufficient 
resistance  in  the  battery  circuit. 

Or  insert  a  sufficient  resistance  in  the  galvano- 
meter circuit,  measure  both,  and  then  measure  the 
added  resistance  alone,  and  subtract  the  lesser  from 
the  greater  resistance ;  the  remainder  will  be  the 
resistance  of  the  galvanometer. 


108 

Or,  as  in  measuring  a  battery  resistance,  after 
obtaining  a  high  deflection,  bring  the  needle  back 
almost  to  zero,  by  placing  a  permanent  magnet 
near  it. 

To  shunt  the  galvanometer,  however,  if  we  have 
a  shunt  convenient,  is  the  preferable  and  most  ele- 
gant way ;  and  when  this  mode  is  adopted,  we  first 
measure  the  resistance  which  is  to  serve  as  the  shunt, 
and  then  loop  it  by  wires  to  the  galvanometer 
terminals. 

Let  us  suppose  the  shunt  measures  100  ohms ; 
and  that  when  in  parallel  circuit  with  the  galvano- 
meter, as  shown  in  Figure  28,  the  deflection  is  re- 
duced to  a  reasonable  figure,  say  30  degrees.  We 


Figure  28. 

find   the  joint  resistance  of  shunt  S,   and  galvan- 
ometer G,  to  be  75  ohms. 


101) 

Now  the  resistance  of  the  shunt  being  100  ohms, 
and  the  joint  resistance  of  both.shunt  and  galvano- 
meter 75  ohms,  how  shall  we,  from  these  figures, 
ascertain  the  galvanometer  resistance  ? 

This  is  so  much  easier  to  explain  by  the  aid  of 
algebra,  that  it  is  not  surprising  that  the  text-book 
writers,  always  drop  into  that  branch  of  mathematics, 
just  as  Silas  Wegg  used  to  drop  into  poetry ;  we 
will,  however,  attempt  to  make  the  thing  plain,  using 
no  other  mathematics  than  arithmetic. 

Resistance  is  the  converse  of  conductivity ;  there- 
fore, if  the  resistance  of  two  wires  or  circuits  are 
known,  their  conducting  power  is  found  by  dividing 
unity  by  their  resistance ;  and  we  say  that  the  con- 
ductivity of  any  wire  is  the  reciprocal  of  its  resist- 
ance. The  reciprocal  of  any  number  is  the  fraction 
obtained  by  dividing  one  by  that  number  ;  and  that 
of  any  fraction  is  the  fraction  itself  inverted.  Thus, 
the  reciprocal  of  4  is  X  or  .25,  and,  conversely,  the 
reciprocal  of  .25,  or  X,  is  f>  which,  of  course,  is 
equivalent  to  4. 

We  are  now  prepared  to  apply  these  principles  to 
the  case  we  have  in  hand,  and,  in  fact,  to  any 
similar  case. 

The  resistance  of  the  shunt  being  100  ohms,  its 
conducting  power  is  the  reciprocal  of  100,  i.  e.y  T^T. 

The  joint  resistance  of  the  galvanometer  and 
shunt  being  75  ohms,  its  conducting  power  is  the 


110 

reciprocal  of  75,  i.  e.,  ?V.  Let  us  call  the  combined 
circuit  G  S  ;  the  shunt  S,  and  the  galvanometer  G. 

It  follows  now,  that  the  conductivity  of  G  S 
being  ?V,  and  that  of  S  alone  T^o,  that  of  G  alone 
must  be  the  difference  between  them,  and  so  it 
proves,  for  subtracting  y^o  from  ^V,  the  remainder 
is  rHo ;  which  by  cancelling  becomes  rro  ;  this  latter 
fraction  representing  the  conducting  power  of  G 
alone. 

We  have  stated  that  the  reciprocal  of  a  fraction 
is  the  fraction  itself  inverted ;  and  inverting  yfo-,  it 
becomes  -f^,  or  300,  which  is  the  required  resistance 
of  the  galvanometer  G. 

Second.  The  second  plan  of  finding  the  resist- 
ance of  the  galvanometer  may  be  adopted  as  an 
alternative  method,  and  is  as  follows : 

Place  the  galvanometer  in  circuit  with  a  rheostat 
or  resistance  box,  and  a  battery  of  very  low  internal 
resistance ;  unplug  any  resistance,  say  400  ohms, 
and  note  the  deflection  :  we  will  suppose  it  to  be  20 
degrees. 

Then  put  the  plugs  back,  withdrawing  resistance 
from  the  circuit,  until  the  former  deflection  is 
doubled,  so  as  to  reach  40  degrees  ;  there  is  now, 
we  may  assume,  300  ohms  unplugged.  We  then 
multiply  the  two  resistances  by  their  respective 
deflections,  subtract  the  smaller  product  from  the 


Ill 

larger,  and  divide  the  result  by  the  difference  be- 
tween the  two  deflections. 

To  make  the  connections  properly  with  the  bridge 
rheostat,  we  connect  the  two  poles  of  the  battery 
with  the  rheostat  terminals  i  and  6,  and  connect 
the  galvanometer  with  terminals  2  and  6.  Leave 
terminals  3,  4  and  5  unattached. 

With  the  figures  we  have  used,  we  first  multiply 
the  400  ohms  by  its  own  deflection  20,  giving  a 
product  of  8,000.  Then  multiplying  300  by  its  own 
deflection  40,  equals  12,000.  12,000  minus  8,000 
leaves  4,000 ;  and  that  amount  divided  by  20,  which 
is  the  difference  between  the  deflections  20  and  40 
degrees,  gives  us  as  the  resistance  of  the  galvan- 
ometer 200  ohms. 

TO    MEASURE    THE    RESISTANCE    OF    THREE    WIRES 
WITHOUT    USING    A    GROUND    WIRE. 

If  we  have  three  line  wires  running  between  the 
same  terminal  stations,  we  can  measure  the  resist- 
ance of  all  of  them  without  an  earth  wire,  and  the 
measurements  will  be  more  correct  than  if  they 
were  measured  as  a  grounded  circuit,  because  errors 
which  arise  from  defective  ground  wires,  and  also 
those  likely  to  accrue  from  earth  currents,  are  thus 
avoided. 

We  may  number  the  wires  i,  2  and  3,  and  we 
desire  to  know  the  resistance  of  each. 


112 

Have  the  distant  ends  of  No.  i  and  No.  2  con- 
nected, and  measure  the  loop.  Let  us  assume  the 
resistance  of  this  loop  to  be  4,000  ohms.  Then  make 
a  second  loop  by  connecting  i  and  3,  at  the  distant 
end ;  which,  when  measured,  is  found  to  have  a 
resistance  of  5,000  ohms. 

Lastly,  let  us  loop  Nos.  2  and  3,  and  measuring 
again  we  find  the  resistance  to  be  8,000  ohms.  Now 
from  these  results  we  have  to  get  the  required  resist- 
ance of  the  three  several  lines. 

To  get  the  resistance  of  No.  i,  we  add  the  first 
two  results  together,  that  is,  the  4,000  and  the  5,000 
ohms ;  the  sum  is  of  course  9,000,  which  is  clearly 
the  sum  of  the  resistance  of  all  the  wires,  the  first 
being  doubled,  as  it  was  measured  both  times. 

We  now  subtract  the  third  result  from  the  sum 
obtained,  deducting  8,000,  the  resistance  of  the  loop, 
composed  of  Nos.  2  and  3,  from  9,000,  and  we  find 
the  remainder  to  be  1,000:  this,  divided  by  2,  because 
No.  i  was  twice  measured,  gives  us  500  ohms  as 
the  resistance  of  No.  i. 

The  resistance  of  No.  2  is  similarly  obtained  ; 
that  is,  by  adding  the  first  and  third  result,  subtract- 
ing the  second,  and  dividing  by  two  ;  and  is  found 
to  be  3,500  ohms.  The  resistance  of  No.  3  is  like- 
wise found  by  adding  the  second  and  third  results, 
subtracting  the  first  from  the  sum  and  dividing  by 
2  ;  leaving  the  final  result  4,500  ohms. 


113 

The  calculation  can  be  expressed   in  figures  as 
follows : 

Resistance  of  No.  i  =  4000+5000 — 8000 


2 
Resistance  of  No.  2=4000  +  8000—500 

• _ 

2 

Resistance  of  No.  3=5000  +  8000  —  4000 


=  500 
=  3500 
=  45°° 


•2 

We  can  of  course  easily  prove  that  these  final 
results  are  correct,  by  adding  them  together. 

TO    MEASURE    THE    INSULATION    RESISTANCE    OF 
A    LINE-WIRE. 

First.  With  the  Bridge.  Use  the  fine  wire  coil 
of  the  galvanometer.  Make  the  connections  pre- 
cisely the  same  as  if  the  wire  resistance  were  to 
be  measured  ;  that  is,  connect  the  battery,  (which 
should  for  insulation  measurements  be  not  less  than 
10  cells),  to  rheostat  terminals  i  and  2  ;  ground  wire 
to  3 ;  line  to  4 ;  and  galvanometer  to  5  and  6. 
Measure  first  with  equal  resistance  in  the  branches. 
If  the  insulation  resistance  is  not  greater  than  the 
total  amount  contained  in  the  rheostat,  zero  can  be 
obtained. 

If  we  find  that  the  needle  will  not  come  to  zero, 
we  now  unplug  unequal  resistances  in  the  branches, 
that  of  F  being  the  greater. 


114 

Let  us,  for  example,  unplug  100  ohms  in  E,  and 
500  in  F,  and  vary  the  plugs  in  the  rheostat,  until 
the  needle  comes  to  zero.  Then,  by  the  usual  pro- 
portion, the  resistance  100  in  E  will  bear  the  same 
ratio  to  the  500,  unplugged  in  F,  as  the  amount 
unplugged  in  the  rheostat  G  doe.s  to  the  insulation 
resistance  measured. 

Suppose  the  line  is  20  miles  long,  and  to  bring 
the  needle  to  zero,  we  have  to  unplug  in  the  rheostat 
3,000  ohms;  then  as  100  is  to  500,  so  3,000  is  to 
15,000,  which  is  the  required  insulation  resistance; 
and  which,  multiplying  by  20,  for  the  number  of 
miles,  gives  a  mileage  insulation  of  300,000  ohms. 

Second.  With  a  tangent  galvanometer.  Find 
first  the  constant  of  the  galvanometer:  that  is,  ascer- 
tain what  deflection  the  galvanometer  gives  with  a 
standard  battery,  through  a  standard  resistance. 
Refer  to  the  table  for  the  tangent  of  the  constant. 
Then  connect  the  galvanometer  in  circuit  with  the 
line  whose  insulation  resistance  is  to  be  measured, 
and  with  the  same  battery  that  was  used  to  take  the 
constant. 

One  pole  of  the  battery  must  be  connected  to 
earth  and.  the  other  to  galvanometer  terminals. 

The  other  galvanometer  terminal  must  be  attached 
to  the  line,  and  the  distant  end  of  the  line  left  open. 
Note  the  deflection.  Find  its  tangent,  by  reference 
to  the  table.  Then  the  tangent  of  the  latter  deflec- 


115 

» 

tion  is  to  the  tangent  of  the  former  as  the  standard 
resistance  is  to  the  insulation  resistance  required. 

For  example :  With  a  tangent  galvanometer, 
connected  in  circuit  with  ten  cells  of  battery  and  a 
standard  resistance  of  5,000  ohms,  we  get  a  deflec- 
tion of  40  degrees.  This  we  call  the  constant. 
Referring  to  the  table  of  tangents,  we  find  the  tan- 
gent of  that  deflection  to  be  .839. 

Then  connect  the  same  galvanometer  and  battery 
to  the  line;  the  resulting  deflection  being,  let 
us  say,  10  degrees;  the  tangent  of  which  is  .176- 
Now,  as  tangent  .176  is  to  tangent  .839,  so  is  5,000 
ohms  to  the  required  insulation  resistance,  23.835 
ohms. 

.839  x  5000 
~7i;6~ 

To  find  the  insulation  resistance  per  mile,  multi- 
ply the  result  of  the  measurement  by  the  number 
of  miles.  Thus  :  if  the  line  is  50  miles  long,  and 
the  insulation  resistance  as  above,  we  multiply 
23>&35  by  50,  and  the  product,  1,191,750  ohms,  is 
the  insulution  resistance  per  mile  ;  which  is  a  very 
fair  grade  of  insulation. 

Insulation  should  never  be  allowed  to  fall  below 
20,0000  ohms  per  mile,  in  wet  weather. 

The  line  whose  insulation  is  to  be  measured, 
must  always  be  open  at  the  distant  end. 


116 


TO    MEASURE,    BY    MEANS    OF    THE    BRIDGE,    THE 

RESISTANCE    OF    A    LINE    WITH    THE    BATTERY 

AT    THE    DISTANT    END. 

In  this  case,  it  is  supposed  that  there  is  no  suita- 
ble battery  at  the  testing  station,  and  that  at  the 
distant  station  there  is  one.  The  formula  is  identi- 
cal in  principle  with  Mance's  method  of  measuring 
the  internal  resistance  of  a  battery ;  and  the  ar- 
rangement of  the  apparatus  is  also  similar. 

Figure  29  illustrates  this  method  of  testing. 


LINE 


Figure  29. 

The  key  only  is  left  in  the  ordinary  place  for  the 
battery.  A  ground  wire  is  attached  to  the  rheostat 
branch  B,  and  the  line  to  the  branch  D. 

In  measuring,  if  the  resistance  of  the  line  is  not 
greater  than  the  total  resistance  in  the  rheostat, 


117 

equal  resistances  may  be  unplugged  in  the  branches 
A  and  C.  If  it  is  greater,  it  will  be  necessary  to 
make  C  greater  than  A.  The  plugs  in  the  rheostat 
branch  B  are  now  to  be  varied  until  the  deflection 
of  the  needle  remains  the  same,  whether  the  key  K 
be  pressed  or  not. 

When  this  occurs,  the  line  is  balanced,  arid  by 
the  usual  process  we  calculate  the  unknown  resist- 
ance, which  will  be  that  of  the  line,  plus  the  internal 
resistance  of  the  battery.  This  ordinarily  can  be 
calculated  and  deducted  from  the  total  measured 
resistance.  The  remainder  is  the  resistance  of  the 
line  alone. 

The  entire  operation  is  simply  that  of  measuring 
the  internal  resistance  of  a  battery  with  very  long 
terminal  wires.  In  practice,  we  connect  the  rheo- 
stat terminal  screws  as  follows  :  Unite  posts  i  and 
2  together  by  a  wire.  Attach  a  ground  wire  to 
post  3  ;  the  line  to  post  4,  and  the  galvanometer  to 
5  and  6. 

Suppose  the  line  to  be  measured  to  have  a  length 
of  300  miles,  and  to  consist  of  No.  9  galvanized 
iron  wire.  To  decide  what  the  branch  resistances 
should  be,  we  may  first  roughly  calculate  the  resist- 
ance of  the  line.  Calling  the  resistance  per  mile  of 
No.  9  galvanized  iron  wire  16  ohms,  the  resistance 
of  the  whole  line  will  be  4,800  ohms. 

Since  this  does  not  exceed  the  total  resistance 


118 

which  may  be  unplugged  in  the  rheostat,  we  may 
use  equal  resistances  in  the  branches. 

We  unplug  500  ohms  in  A  and  in  C,  and  change 
the  plugs  in  the  rheostat,  finding,  perhaps,  that  the 
deflection  remains  constant  when  we  have  un- 
plugged 5,100  ohms. 

Then  5,100  ohms  is  the  resistance  of  the  line  and 
battery.  We  find  that  the  battery  consists  of  eighty 
Callaud  cells,  and  assuming  one  Callaud  cell  to  have 
a  resistance  of  2  ohms,  which  gives  a  total  of  160 
ohms  for  the  whole  battery,  we  deduct  that  amount 
(160)  from  the  5,100,  which  leaves  us,  as  the  resist- 
ance of  the  line,  4,940  ohms. 

As  in  measuring  the  internal  resistance  of  a  bat- 
tery by  the  Mance  method,  we  shall  probably  have 
to  reduce  the  deflection  by  one  of  the  plans  des- 
cribed in  the  explanation  of  that  method,  or  by 
reducing  the  battery  at  the  distant  station. 

TO    MEASURE    A    LINE    WITHOUT    A    BATTERY. 

Occasionally,  earth  currents  are  so  strong  on  a 
line,  as  to  make  it  impossible  to  make  an  accurate 
measurement.  In  this  case,  if  the  earth  current  is 
of  constant  direction,  the  best  way  to  proceed  is  to 
disconnect  the  battery  altogether,  and  then,  regard- 
ing the  earth  current  as  the  battery  power,  proceed 
as  in  the  plan  just  described  of  measuring  the  line, 
using  a  distant  battery. 


TO    COMPARE,    MEASURE,   OR   ESTIMATE   THE    ELECTRO- 
MOTIVE   FORCE    OF    A    BATTERY. 

There  is  no  absolute  permanent  standard  of  elec- 
tro-motive force,  because  no  battery  can  stay  in  an 
absolutely  constant  condition,  and  therefore  we 
cannot  arbitrarily  determine  the  force  of  any  par- 
ticular battery  in  standard  units  (volts),  but  can 
only  compare  a  battery  whose  electro-motive  force 
is  not  known,  with  one  whose  force  we  know. 
This  comparison  may  be  made  in  several  ways. 

We  will  describe  one  or  two  of  the  easiest  and 
plainest  methods  in  use. 

First :  Join  up  a  number  of  the  cells  whose 
electro-motive  force  we  desire  to  know,  in  circuit 
with  a  galvanometer,  and  also  with  a  number  of 
cells  whose  electro-motive  force  we  do  know — the 
latter  being  connected  in  opposition  to  the  former, 
—then  adjust  the  number  of  cells  of  each  series,  so 
that  one  series  just  balances  the  other,  and  no  cur- 
rent passes.  When  this  point  is  reached,  the  needle 
has  no  deflection,  and  the  relative  force  of  the 
batteries  may  thus  be  determined.  For  example  : 
We  desire  to  know  the  electro-motive  force  of  a 
bichromate  battery  of  10  cells,  and  we  have  a 
Daniell  battery  with  which  we  can  compare  it.  We 
know  that  a  Daniell  cell  in  good  order  is  about  one 
volt ;  in  exact  terms,  1.079  v°lts.  We  connect 


120 

one  pole — the  zinc,  for  instance — of  our  bichromate 
battery,  to  one  terminal  of  the  galvanometer,  and 
the  carbon  pole  to  the  copper  pole  of  a  battery 
composed  of  an  equal  number  of  Daniell  cells  ;  the 
zinc  pole  of  the  Daniell  battery  is  connected  to  the 
other  terminal  of  the  galvanometer.  We  see  that 
the  needle  deflects.  One  of  the  batteries  is  evid- 
ently stronger  than  the  other.  We  add  another 
cell  of  the  bichromate  battery,  and  the  deflection 
increases  ;  this  shows  that  we  are  on  the  wrong 
tack,  and  we  reduce  the  bichromate  battery  to  its 
original  number,  and  add  cells  to  the  Daniell  in- 
stead, until  the  needle  deflects  no  longer.  Suppose 
we  have  to  add  6  cells  to  bring  the  needle  to  zero. 
It  thus  takes  16  Daniell  cells  to  balance  10  bi- 
chromate cells,  showing  that  the  E.  M.  F.  of  the 
bichromate  battery  is  to  that  of  the  Daniell  as  16  is 
to  10,  or  cancelling,  as  8  is  to  5. 

To  find  the  value  in  volts,  multiply  the  E.  M.  F. 
of  one  Daniell  cell  1.079,  by  the  number  of  cells  16, 
and  divide  by  the  number  of  bichromate  cells  10. 
The  quotient  is  1.726,  which  is  the  value  of  bichro- 
mate cell. 

This  plan  can  be  adopted  with  the  galvanometer 
of  the  bridge  set. 

Second :  With  a  tangent  galvanometer.  The 
electro-motive  forces  of  two  batteries  are  to  be  com- 
pared. Call  them  A  and  B.  Connect  up  A  in 


121 

circuit  with  a  galvanometer  and  rheostat.  Unplug 
sufficient  resistance  to  produce  a  convenient  deflec- 
tion. Observe  the  tangent  of  the  deflection,  and 
note  also  the  total  resistance  in  circuit :  that  is,  the 
sum  of  the  resistances  of  the  battery  galvanometer, 
and  whatever  is  unplugged  in  the  rheostat. 

Now  remove  battery  A,  and  substitute  battery  B. 
If  the  internal  resistance  of  B  is  different  from  A, 
the  resistance  unplugged  must  be  changed,  and  re- 
adjusted, until  the  total  resistance  in  circuit  is  the 
same  as  before.  Again  note  the  tangent  of  the 
deflection. 

The  electro-motive  force  of  A  is  now  to  the 
electro-motive  force  of  B,  as  the  first  tangent  ob- 
served is  to  the  second. 

For  example  :  We  have  two  batteries,  each  of 
which  is  composed  of  20  cells.  We  call  the  first 
battery  A,  and  its  electro-motive  force  is  20  volts. 
We  wish  to  determine  the  E.  M.  F.  of  the  second 
battery  which  we  may  call  B. 

Suppose  A  to  have  a  resistance  of  60  ohms,  and 
the  galvanometer  100  ohms.  We  unplug  800  ohms 
in  the  rheostat,  making  a  total  resistance  of  960 
ohms.  With  this  resistance  we  find  that  the  needle 
deflects  to  35  degrees.  Referring  to  the  table  of 
tangents,  we  find  that  the  tangent  of  35  is  .70. 

We  note  the  above  facts,  and  then  disconnect 
battery  A,  putting  battery  B,  which  has  a  resistance 


122 

of  ipo  ohms,  in  its  place.  We  must  now  reduce 
the  resistance  unplugged  in  the  rheostat,  to  760  ohms, 
so  as  to  make  the  total  resistance  the  same  as  before, 
i.  e.y  960  ohms. 

With  this  resistance  suppose  we  now  get  a  deflec- 
tion of  42  degrees,  the  tangent  of  which  is  .90  ;  then 
as  .70  is  to  .90,  so  is  the  E.  M.  F.  of  A,  to  the  E.  M. 
F.  of  B ;  and  since  the  E.  M.  F.  of  A  was  20  volts, 
the  calculation  is  that  .70  is  to  .90,  as  20  volts  is  to 
256-  volts. 

A  third  method  consists  in  placing  each  battery 
alternately  in  circuit,  varying  resistance  to  produce 
the  same  deflection  with  each,  then  adding  the  re- 
quired resistance  in  both  cases,  to  produce  lower, 
but  again  similar  deflections;  the  electro-motive 
forces  then  being  directly  proportional  to  the  added 
resistances,  which  in  both  cases  were  required. 

To  illustrate  :  No  i  battery,  which  we  will  sup- 
pose has  a  known  E.  M.  F.  of  25  volts,  is  placed  in 
circuit  with  a  galvanometer  and  rheostat.  Unplug 
say  2,000  ohms,  and  note  the  deflection,  which  is  30 
degrees.  Adding  200  ohms  to  that  already  unplug- 
ged, brings  the  deflection  down  to  24  degrees  Dis- 
connect No.  i  and  connect  No.  2  in  its  place.  We 
find  that  to  produce  the  same  deflection  as  the  one 
we  had  first — 30  degrees,  we  have  to  unplug  but 
i, 800  ohms ;  and  by  adding  150  ohms,  we  bring  the 


123 
- 

deflection  down  to  that  produced  by  adding  when 
No.  i  was  in  the  circuit,  24  degrees. 

Now  the  amount  added  in  the  measurement  of 
No.  i,  that  is  200  ohms,  is  to  the  amount  added  in 
the  measurement  of  No.  2,  viz.,  150  ohms,  as  the 
E.  M.  F.  of  No.  i,  that  is  25  volts,  is  to  i&K  volts, 
the  E.  M.  F.  of  No.  2. 

It  is  a  good  idea  to  keep  a  record  of  the  deflection 
which,  with  a  given  battery,  having  a  known  E.  M. 
F.,  and  given  resistance,  will  be  produced  upon  a 
given  galvanometer,  because,  by  so  doing,  we  can 
always  with  the  same  galvanometer  use  that  deflec- 
tion as  a  standard. 


TO    LOCATE    A    BAD    JOINT,    OR    OTHER    DEFECTIVE 

POINT,    PRODUCING    A    HIGH    RESISTANCE 

IN    A    LINE    CIRCUIT. 

A  line  circuit,  whose  resistance,  when  in  good 
order,  should  be  (as  known  from  prior  measure- 
ments, or  by  calculation),  for  example,  about  3,000 
ohms,  is  found  to  require  a  much  larger  battery 
power  than  it  ought,  to  produce  a  current  of  required 
strength,  and  on  being  tested,  shows  a  resistance  of 
say  10,000  ohms  :  the  cause  is  probably  a  bad  or 
unsoldered  joint ;  although  it  may  be  a  few  loose 
instrument  connections,  or  a  poor  terminal  ground. 


124 

The  defect  may  be  localized  by  the  following  pro- 
cess : 

Attach  a  ground  wire  to  the  middle  of  the  line, 
and  with  the  distant  end  opened,  measure  to  the  mid- 
dle. If  the  bulk  of  the  resistance  is  still  in  the  half 
of  the  line  measured,  the  defect  is  between  the  mid- 
dle of  the  line  and  the  testing  station  ;  now  have  the 
ground  wire  moved  to  the  middle  of  the  half  measur- 
ed, and  make  a.third  test,  and  so  on  until  the  trouble 
is  passed,  and  the  measured  resistance  makes  a  sud- 
den fall  from  the  last  one  ;  when  this  occurs,  the 
defect  is  located  between  the  last  two  measurements. 

If,  on  the  contrary,  on  making  the  first  test  with 
the  middle  of  the  line  grounded,  the  resistance  falls 
to  such  an  amount  as  half  of  the  line  would  norm- 
ally possess,  the  defect  is  evidently  in  the  open  half 
of  the  line,  beyond  the  middle  ground  wire,  and 
must  be  followed  up  until  it  appears  on  the  testing 
side  of  the  temporary  ground  once  more.  Then  it 
is  located  between  the  two  last  temporary  terminals. 


TO    TEST    FOR    A    GROUND    WITH    THE    BRIDGE. 

Have  the  grounded  wire  looped  to  a  perfect  wire 
at  the  distant  station  ;  connect  the  loop  to  rheostat 
posts  3  and  4,  and  measure  it :  call  the  resistance 
4,500  ohms.  Connect  a  ground  wire  to  post  3,  and 


125 

the  perfect  wire  to  post  4,  and  measure  it,  the  dis- 
tand  end  being  grounded  at  the  fault,  and  the  defec- 
tive wire  open  at  the  testing  end.  Call  the  resistance 
3,050  ohms  ;  then  connect  the  defective  wire  to 
post  4,  and  measure  to  the  fault,  leaving  the  ground 
wire  in  post  3,  and  the  good  wire  open.  Call  resist- 
ance 1,510  ohms. 

Thus  we  have  now  three  measurements;  viz.,  the 
metallic  loop,  4,500  ohms  ;  the  long  end  of  the  loop 
measured  to  ground  at  the  fault,  3,050  ohms,  and 
the  short  end,  measured  in  the  same  way,  1,510 
ohms.  We  might  at  first  suppose  that  the  two  sides 
of  the  loop  measured  separately  would,  added  to- 
gether, give  a  sum  equal  to  the  resistance  of  the 
loop  ;  but  we  see  this  is  not  so  ;  that  the  sum  of  the 
two  measurements  is  4,560  ohms,  or  sixty  ohms 
more  than  the  resistance  of  the  loop. 

This  surplus  of  60  ohms,  is  evidently  the  resist- 
ance of  the  fault  itself,  measured  twice  over ;  once 
when  we  measured  the  long  side  of  loop,  and  once 
when  we  measured  the  short  side,  and  the  true  re- 
sistance of  the  fault  is  therefore  30  ohms. 

To  ascertain  this,  we  add  the  two  last  measure- 
ments together,  and  from  their  sum  deduct  the 
first,  dividing  the. result  by  2  ;  the  final  result  being 
30  ohms,  which,  as  before  stated,  is  the  resistance 
of  the  fault. 

Then  the  resistance  between  the  testing  station 


126 

and  the  fault  is  found,  by  deducting  the  resistance 
of  the  fault,  from  the  measurement  of  the  defective 
wire  to  ground  at  the  fault ;  that  is,  deduct  30  from 
1,510,  leaving  a  line  resistance  of  1,480  ohms  be- 
tween the  testing  station  and  the  ground. 

If  the  ordinary  resistance  of  the  entire  line,  the 
number  of  stations,  and  the  length  of  the  line  in 
miles  are  known,  it  is  easy  from  the  foregoing  data, 
to  calculate  the  approximate  location  of  the  ground'. 

Suppose  each  of  the  two  lines  to  have  in  circuit, 
5  relays  of  150  ohms  resistance,  making  a  total  re- 
sistance, when  in  good  working  order,  as  the  loop 
measurement  shows,  of  2,250  ohms.  Suppose  also 
that  the  line  is  100  miles  long :  this  gives  a  mileage 
resistance  of  22^  ohms,  including  the  relays. 

Then  the  distance  between  the  testing  station 
and  the  fault,  is  65-^  miles ;  which,  of  course  is  de- 
termined by  dividing  the  line  resistance  to  the  fault, 
1,480  ohms,  by  the  resistance  per  mile. 

Such  tests  as  these  will  be  more  simple,  as  well 
as  more  accurate,  if  the  relays  can  be  cut  out  of 
circuit  before  commencing  to  test. 


TO    TEST    FOR    A    CROSS    WITH    A    GALVANOMETER. 

First.   When  the  two  crossed  wires  are  very  nearly 
identical  in  resistance. 


127 


As  in  the  diagram,  Figure  30,  call  No.  i  A,  B  : 
No.  2,  C,  D,  and  the  cross  X.  We  measure  No.  i, 
from  A  to  B,  leaving  No.  2  open  at  C  and  D,  find- 
ing the  resistance  to  be,  say  2,000  ohms. 


Figure  30. 

Measuring  A  to  D  through  the  cross,  leaving  C 
and  B  open,  we  find  this  resistance  2,150  ohms. 
Then  measuring  A  to  C  as  a  loop,  leaving  B  and 
D  open,  we  find  the  loop  to  have  a  resistance  of 
3,550  ohms. 

The  resistance  of  the  cross  itself  is  found  by  sub- 
tracting the  result  of  the  first  measurement  from 
that  of  the  second;  that  is  2,000  from  2,150,  and 
is  thus  150  ohms:  and  the  resistance  from  the  test- 
ing station  A  to  the  cross  X  on  No.  i,  is  now  de- 
termined by  deducting  the  resistance  of  the  cross, 
i.  e.,  150  ohms,  from  the  loop  measurement  3,550 
ohms,  and  dividing  the  result  by  2  ;  giving  as  the 
required  resistance  1,700  ohms. 

This  may  be  reduced  to  miles,  in  the  manner  ex- 
explained,  in  the  test  for  grounds. 

Suppose  the  lines  are  100  miles  long:  since  we 
found  by  measuring  from  A  to  B  that  the  resistance 
of  No.  i  with  No.  2  open,  was  2,000  ohms;  it  is 


128 

clear  that  2,000  ohms  is  to  100  miles,  as  1,700  ohms 
is  to  85  miles,  which  is  the  distance  of  the  cross  X 
from  the  testing  station  A. 

Second. — To  test  for  a  cross  when  the  resistance 
of  the  wires  are  not  alike. 

We  have  two  wires  crossed,  No.  i  and  No.  2,  as 
shown  in  Figure  31. 

No.i.4 z JL & 5 

No.2.0. 


Both  wires  have  the  same  termini,  but  have  differ- 
ent resistances.  We  desire  to  locate  the  cross  X. 

No.  i  we  will  again  suppose  to  be  100  miles  long. 
We  proceed  as  follows  : 

Measure  No.  i,  A  to  B,  with  C  and  D  open  :  call 
it  2,000  ohms.  This  is  the  regular  resistance  of 
No.  i. 

Measure  C  to  B,  through  the  cross,  A  and  D  being 
left  open,  call  it  2,750  ohms. 

Measure  now  the  loop  from  A,  through  the  cross, 
and  back  to  C,  leaving  B  and  D  open ;  call  it  3,100 
ohms. 

Call  No.  i  on  the  near  side  of  the  cross  Z,  and  on 
the  far  side  K.  Call  also  No.  2  on  the  near  side  of 


129 

the  cross  Y,  and  the  resistance  of  the  cross  itself  X. 

Now,  by  examining  the  diagram,  we  may  see  that 
we  have  made  measurements  as  follows :  i  st,  from 
A  to  B,  2,000  ohms,  that  is,  the  sum  of  the  two 
sections  Z  and  K  ;  2d,  from  C  to  B,  2,750  ohms, 
that  is,  the  sum  of  Y,  K  and  X  ;  3d,  from  A  to  C, 
3,100  ohms,  that  is,  the  sum  of  Z,  Y  and  X. 

We  now  add  the  results  of  the  first  and  third 
measurements,  2,000  and  3,100  ohms,  together,  the 
sum  consisting  of  the  following  elements  :  Z  meas- 
ured twice,  Y  once,  X  once,  and  K  once  ;  or,  No.  i 
from  A  to  the  cross  twice,  No.  2  from  C  to  the 
cross  once,  the  cross  itself  once,  and  No.  i  from 
the  cross  to  B  once. 

Next  we  subtract  the  result  of  the  second  measure- 
ment, 2,750  ohms,  which  consists,  as  we  have  seen, 
of  Y,  X  and  K,  each  measured  once,  that  is,  No.  2 
from  C  to  the  cross  once,  the  cross  itself  once,  and 
No.  i  from  the  cross  to  B  once,  from  the  sum, 
5,100  ohms,  of  the  first  and  third ;  the  remainder  is, 
of  course,  2,350  ohms,  which  represents  that  part 
of  No.  i  which  we  have  callen  Z  doubled  ;  dividing 
that  number  by  2,  we  find  the  resistance  of  Z  to  be 
1,175  ohms. 

Then,  as  the  length  of  the  line  is  100  miles,  and 
the  entire  resistance  from  A  to  B,  that  is,  the  sum 
of  Z  and  K,  2,000  ohms,  it  is  clear  that  the  resistance 
of  Z,  or  the  portion  of  the  line  between  A  and  X, 


130 

must  be  58!  miles,  for  as  2,000  ohms  is  to  100  miles, 
so  1,175  ohms  is  to  58f  miles,  the  distance  of  the 
cross  from  A. 

TO    TEST    A    TERMINAL    GROUND    AT    A    DISTANT 
STATION. 

Measure  any  two  lines  to  earth  through  the  same 
ground  wire. 

Then  measure  the  two  wires  looped  together  at 
the  distant  station. 

Add  together  the  resistances  of  the  two  lines 
when  measured  separately,  and  compare  with  the 
resistance  of  the  loop. 

If  the  resistance  of  the  sum  of  the  two  is  greater 
than  that  of  the  loop,  the  ground  wire  is  not  perfect, 
and  its  resistance  is  found  by  deducting  the  resist- 
ance of  the  loop  from  that  of  the  sum  of  the  separate 
measurements,  and  dividing  the  remainder  by  2. 

For  example  :  We  are  at  a  testing  station  A  ;  we 
desire  to  test  the  terminal  ground  at  B.  We  meas- 
ure No.  i  to  ground  at  B,  and  find  its  resistance  to 
measure  2,500  ohms. 

Measuring  No.   2,  in   the  same  way,  we  get  a 
measured  resistance  of  3,600  ohms. 
.     Disconnecting  both  wires  from  the  distant  ground, 
and  connecting  them  as  a  loop,  we  measure  the  loop, 
and  find  the  result  to  be  5,950  ohms. 


131 

The  sum  of  the  first  two  measurements,  2,500 
and  3,600  ohms,  is  6, 100  ohms,  which,  as  we  see,  is 
150  ohms  greater  than  the  resistance. of  the  loop. 

The  ground  wire  is,  therefore,  defective.  To 
ascertain  its  resistance,  we  divide  the  remainder 
150  by  2,  which  shows  the  resistance  of  the  ground 
wire  to  be  75  ohms. 

In  all  of  the  foregoing  tests,  when  we  measure  a 
loop,  the  two  wires  of  the  loop  are  to  be  connected 
with  the  rheostat  binding  screws  3  and  4.  When  we 
measure  a  single  line,  we  connect  a  ground  wire  with 
post  3,  and  the  line  with  post  4.  The  battery  and 
galvanometer  are  connected  in  their  usual  places. 

TO    MEASURE    THE    RESISTANCE    OF    AN    ELECTRIC 
LIGHT. 

Since  it  is  often  necessary  to  ascertain  the  resist- 
ance of  an  electric  lamp  while  burning,  it  has  been 
thought  well  to  refer  in  conclusion  to  that  subject. 
The  measurement  can  be  readily  made  by  the 
Wheatstone  Bridge  apparatus,  or  by  the  use  of  a 
tangent  galvanometer,  although,  of  course,  the  fine 
instruments  we  have  described  are  not  adapted  for 
use  with  the  strong  currents  requisite  in  electric 
lighting. 

When  we  deal  with  such  currents,  we  have  to 
employ  galvanometers  of  very  low  resistance,  and 


132 

rheostats  and  resistance  coils  composed  of  very  large 
wire,  and  in  some  cases  it  is  convenient  to  use 
resistances  of  other  material,  such  as  carbon. 

Figure  32  shows  a  theoretical  arrangement  for 
measuring  the  resistance  of  an  electric  lamp. 

The  wires  i  and  2  lead  from  a  suitable  dynamo- 
electric  machine  M  to  the  Bridge  terminals. 

In  the  branches  A  and  C  suitable  resistances  are 
placed  as  usual ;  the  principal  adjustable  resistance 
is  included  in  the  branch  B,  and  the  lamp  to  be 
measured  in  D. 


Figure  32. 

It  will  generally  be  best  to  make  the  resistance  in 
A  larger  than  that  in  C,  so  that  B  will  also  have  to 
be  proportionately  larger  than  the  resistance  of  the 
lamp  in  D. 


133 

We  thus  cause  the  larger  part  of  the  current  to 
pass  through  the  branches  C  and  D,  and  afford  the 
necessary  supply  of  electricity  to  the  lamp.  Instead 
of  first  using  a  galvanometer,  it  has  been  found 
advantageous  to  insert  a  relay  R  in  the  cross  wire, 
until  the  balance  is  so  nearly  established,  that  the 
magnet  doqs  not  seem  to  be  affected.  We  can  then 
safely  substitute  even  a  fine  wire  galvanometer,  and 
conclude  the  measurement  more  accurately,  calcu- 
lating the  result  by  proportion  in  the  same  way  as  we 
would  in  the  measurement  of  any  other  resistance. 

By  using  a  polarized  relay,  we  not  only  detect  the 
passage  of  a  current  in  the  cross  wire,  but  also  its 
direction. 


134 


TABLE    OF   TANGENTS. 


Degrees. 

Tangents. 

Degrees. 

Tangents. 

Degrees. 

Tangents. 

Degrees. 

Tangents. 

I. 

.OI/S 

13. 

.2309 

25. 

.4663 

37- 

.7536 

i-5 

.0262 

13.5 

.2401 

25.5 

.4770 

37-5 

.7673 

2. 

•0349 

14. 

•2493 

26. 

4877 

38. 

.7813 

2-5 

•0437 

H-5 

.2586 

26.5 

.4986 

38.5 

•7954 

3- 

.0524 

IS- 

.2679 

27. 

-5095 

39- 

.8098 

3-5 

.0612 

15-5 

.2773 

27.5 

5206 

39-5 

.8243 

4- 

.0699 

16. 

.2867 

28. 

.5317 

40. 

.8391 

4-5 

.0787 

16.5 

.2962 

28.5 

5430 

40.5 

.8541 

5- 

.0875 

17- 

•3°57 

29. 

•5543 

41. 

.8693 

5.5 

.0963 

17.5 

.3'53 

29.5 

.5658 

4i.5 

.8847 

6. 

.  1051 

18. 

.3249 

30. 

•5774 

42. 

.9004 

6-5 

•1139 

18.5 

.3346 

30.5 

.5890 

42.5 

.9163 

7- 

.1228 

19. 

•  3443 

31. 

.6009 

43- 

.9325 

7-5 

.1317 

J9-5 

•  3541 

31-5 

.6128 

43-5 

.9490 

8. 

.1405 

20. 

.3640 

32. 

.6249 

44- 

.9657 

8.5 

•1495 

20.5 

•  3739 

32-5 

•6371 

44-5 

•9827 

9- 

.1584 

21  . 

.3839 

33- 

.6494 

45- 

i. 

9-5 

.1673 

21.5 

•3939 

33  5 

.6619 

45-5 

.0176 

10. 

.1763 

22. 

.4040 

34- 

.6745 

46. 

•  035 

10.5 

•1853 

22.5 

.4142 

34-5 

6873 

46.5 

•  053 

u. 

.1944 

23. 

.4245 

35- 

.7002 

47- 

.072 

11.5 

.2035 

23.5 

•4348 

35-5 

.7133 

47-5 

.091 

12. 

.2126 

24. 

.4452 

36. 

.7265 

48. 

.  IIO 

12.5 

.2217 

24.5 

•4557 

36.5 

7400 

48.5 

.130 

135 


TABLE  OF  TANGENTS — Continued. 


Degrees. 

Tangents. 

Degrees. 

Tangents. 

Degrees. 

Tangents. 

Degrees. 

Tangents. 

49- 

I.  ISO 

59-5 

.697 

70. 

2-747 

80.5 

5-975 

49-5 

I.  170 

60. 

.732 

70.5 

2.823 

81. 

6.313 

50. 

I  .  IQI 

60.5 

.767 

71. 

2.904 

81.5 

6.691 

50.5 

I.2I3 

61. 

.804 

7L5 

2.988 

82. 

7.115 

5i. 

1-234 

61.5 

.841 

72. 

3.077 

82.5 

7-595 

5L5 

1.257 

62. 

.880 

72.5 

3.I7I 

S3- 

8.144 

52. 

1.279 

62.5 

.921 

73. 

3.270 

83-5 

8.776 

52.5 

1.303 

63. 

.962 

73-5 

3.375 

84. 

Q-SH 

53- 

1.327 

63.5 

2.005 

74. 

3.487 

84.5 

10.385 

53-5 

L35I 

64- 

2.050 

74.5 

3.605 

85. 

11.430 

54- 

1.376 

64.5 

2.O96 

75- 

3-732 

85.5 

12.706 

54-5 

I.40I 

65. 

2.144 

75-5 

3.866 

86. 

14.300 

55- 

1.428 

65.5 

2.194 

76. 

4.OIO 

86.5 

16.349 

55-5 

1-455 

66. 

2.246 

76.5 

4.165 

87. 

19.081 

56. 

1.482 

66.5 

2.299 

77- 

4.331 

87.5 

22.903 

56.5 

I.5IO 

67. 

2.355 

77-5 

4.510 

88. 

28.636 

57- 

L539 

67.5 

2.414 

78. 

4.704 

88.5 

38.188 

57-5 

1.569 

68. 

2-475 

78.5 

4.915 

89. 

57.290 

58. 

I.  6OO 

68.5 

2.538 

79- 

5-144 

89.5 

114.588 

58.5 

I  .631 

69. 

2.605 

79-5 

5-395 

90. 

59- 

1.664 

69.5 

2.674  ; 

80. 

5.671 

136 


TABLE   OF   SINES. 


Degrees. 

Sines. 

Degrees. 

Sines. 

Degrees. 

Sines. 

Degrees. 

Sines. 

I  . 

.0175 

12. 

.2079 

23-    .3907 

34- 

•5592 

I.? 

.0262 

12.5 

.2164 

23.5  '.3987 

34-5 

.5664 

2. 

•0349 

13- 

.2250 

24. 

.4067 

35- 

5736 

2-5 

.0436 

13.5 

-2334 

24.5 

.4147 

35-5 

.5807 

3- 

.0523 

H. 

.2419 

25. 

.4226 

36. 

.5878 

3-5 

.0610 

H.5 

.2504 

25.5 

.4305 

36.5 

.5948 

4- 

.0698 

15. 

.2588 

26. 

.4384 

37- 

.6018 

4-5 

.0785 

15.5 

2672 

26.5 

.4462 

37-5 

.6688 

5- 

.0872 

16. 

.2756 

27. 

.4540 

38. 

.6157 

5-5 

.0958 

I6.5 

.2840 

27.5 

.4617 

38.5 

.6225 

6. 

.1045 

17. 

.2924 

28 

.4695 

39- 

.6293 

6.5 

.1132 

17-5 

.3007 

28.5 

.4772 

39-5 

.6361 

7- 

.  1219 

18. 

.3090 

29. 

.4848 

40. 

.6428 

7-5 

.1303 

18.5 

.3173 

29.5 

.4924 

40.5 

.6494 

8. 

.1392 

19. 

.3256 

30. 

.5000 

41. 

.6561 

8.5 

.1478 

19.5 

•3338 

30-5 

5075 

41.5 

.6626 

9- 

.1564 

20 

.3420 

31. 

.5150 

42. 

.6691 

9  5 

.1650 

20.5 

.3502 

3L5 

.5225 

42.5 

.6756 

10. 

.1736 

21  . 

3584 

32. 

.5299 

43- 

.6820 

10.5 

.1822 

21.5 

.3665 

32.5 

•5373 

43-5 

.6884 

ii  . 

.1908 

22. 

.3746 

33- 

.5466 

44- 

.6947 

u.  5 

.1994 

22.5 

.3827 

33-5 

•55i9 

44-5  -7009 

137 


TABLE  OF  SINES — Continued. 


Degrees. 

Sines. 

Degrees. 

Sines. 

Degrees. 

Sines. 

Degrees. 

Sines. 

45- 

.7071 

56.5 

•8339 

68. 

.9272 

79-5 

-9833 

45-5 

•7133 

57- 

8387 

68.5 

.9304 

80. 

.9848 

46. 

7193 

57-5 

•8434 

69. 

9336 

80.5 

.9863 

46.5 

•7254 

58. 

.8480 

69.5 

9367 

81. 

.9877 

47- 

•73H 

58.5 

.8526 

70. 

•9397 

81.5 

.9890 

47-5 

•7373 

59- 

.8572 

70.5 

.9426 

82. 

.9903 

48. 

•7431 

59-5 

.86l6 

7i. 

9455 

82.5 

.9914 

48.5 

749° 

60. 

.8660 

71-5 

.9483 

83. 

.9925 

49- 

•7547 

60.5 

.8704 

72. 

.9511 

83.5 

.9936 

49-5 

.7604 

61. 

.8746 

72.5 

•9537 

84. 

•9945 

50. 

.7660 

61.5 

.8788 

73- 

.9563 

84.5 

•9954 

50.5 

.7716 

62. 

.8829 

73-5 

•9588 

85. 

.9962 

51. 

.7771 

62.5 

.8870 

74- 

•9613 

85.5 

.9969 

5i.5 

.7826 

63. 

.8910 

74-5 

.9636 

86. 

.9976 

52. 

.7880 

63.5 

.8949 

75- 

.9659 

86.5 

.9981 

52.5 

•7934 

64. 

.8988 

75-5 

.9681 

87.    .9986 

53- 

.7986 

64-5 

9026 

76. 

.9703 

87.5 

.9990 

53-5- 

.8039 

65- 

9063 

76  5 

.9724 

88.  |  .9994 

54- 

.8090 

65.5 

.9IOO 

77- 

•9744 

88.5   .9997 

54-5 

.8141 

66. 

.9135 

77-5 

9763 

89. 

.9998 

55- 

.8192 

66.5 

.9171 

78. 

.9781 

89.5 

i  . 

55-5 

.8241 

67. 

.9205 

78.5 

9799 

90. 

i. 

56. 

.8290 

67.5 

.9239 

79- 

.9816 

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